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Solving integrals/ cos^27x

Integral of cos^27x dx

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∫2∞cos⁡27(x) dx\int\limits_{2}^{\infty} \cos^{27}{\left(x \right)}\, dx 
Integral(cos(x)^27, (x, 2, oo))
Detail solution

  1. Rewrite the integrand:

    cos⁡27(x)=(1−sin⁡2(x))13cos⁡(x)\cos^{27}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{13} \cos{\left(x \right)}

  2. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      (1−sin⁡2(x))13cos⁡(x)=−sin⁡26(x)cos⁡(x)+13sin⁡24(x)cos⁡(x)−78sin⁡22(x)cos⁡(x)+286sin⁡20(x)cos⁡(x)−715sin⁡18(x)cos⁡(x)+1287sin⁡16(x)cos⁡(x)−1716sin⁡14(x)cos⁡(x)+1716sin⁡12(x)cos⁡(x)−1287sin⁡10(x)cos⁡(x)+715sin⁡8(x)cos⁡(x)−286sin⁡6(x)cos⁡(x)+78sin⁡4(x)cos⁡(x)−13sin⁡2(x)cos⁡(x)+cos⁡(x)\left(1 - \sin^{2}{\left(x \right)}\right)^{13} \cos{\left(x \right)} = - \sin^{26}{\left(x \right)} \cos{\left(x \right)} + 13 \sin^{24}{\left(x \right)} \cos{\left(x \right)} - 78 \sin^{22}{\left(x \right)} \cos{\left(x \right)} + 286 \sin^{20}{\left(x \right)} \cos{\left(x \right)} - 715 \sin^{18}{\left(x \right)} \cos{\left(x \right)} + 1287 \sin^{16}{\left(x \right)} \cos{\left(x \right)} - 1716 \sin^{14}{\left(x \right)} \cos{\left(x \right)} + 1716 \sin^{12}{\left(x \right)} \cos{\left(x \right)} - 1287 \sin^{10}{\left(x \right)} \cos{\left(x \right)} + 715 \sin^{8}{\left(x \right)} \cos{\left(x \right)} - 286 \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 78 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 13 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}

    2. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−sin⁡26(x)cos⁡(x)) dx=−∫sin⁡26(x)cos⁡(x) dx\int \left(- \sin^{26}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{26}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u26 du\int u^{26}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u26 du=u2727\int u^{26}\, du = \frac{u^{27}}{27}

          Now substitute uu back in:

          sin⁡27(x)27\frac{\sin^{27}{\left(x \right)}}{27}

        So, the result is: −sin⁡27(x)27- \frac{\sin^{27}{\left(x \right)}}{27}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫13sin⁡24(x)cos⁡(x) dx=13∫sin⁡24(x)cos⁡(x) dx\int 13 \sin^{24}{\left(x \right)} \cos{\left(x \right)}\, dx = 13 \int \sin^{24}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u24 du\int u^{24}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u24 du=u2525\int u^{24}\, du = \frac{u^{25}}{25}

          Now substitute uu back in:

          sin⁡25(x)25\frac{\sin^{25}{\left(x \right)}}{25}

        So, the result is: 13sin⁡25(x)25\frac{13 \sin^{25}{\left(x \right)}}{25}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−78sin⁡22(x)cos⁡(x)) dx=−78∫sin⁡22(x)cos⁡(x) dx\int \left(- 78 \sin^{22}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 78 \int \sin^{22}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u22 du\int u^{22}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u22 du=u2323\int u^{22}\, du = \frac{u^{23}}{23}

          Now substitute uu back in:

          sin⁡23(x)23\frac{\sin^{23}{\left(x \right)}}{23}

        So, the result is: −78sin⁡23(x)23- \frac{78 \sin^{23}{\left(x \right)}}{23}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫286sin⁡20(x)cos⁡(x) dx=286∫sin⁡20(x)cos⁡(x) dx\int 286 \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx = 286 \int \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u20 du\int u^{20}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u20 du=u2121\int u^{20}\, du = \frac{u^{21}}{21}

          Now substitute uu back in:

          sin⁡21(x)21\frac{\sin^{21}{\left(x \right)}}{21}

        So, the result is: 286sin⁡21(x)21\frac{286 \sin^{21}{\left(x \right)}}{21}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−715sin⁡18(x)cos⁡(x)) dx=−715∫sin⁡18(x)cos⁡(x) dx\int \left(- 715 \sin^{18}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 715 \int \sin^{18}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u18 du\int u^{18}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u18 du=u1919\int u^{18}\, du = \frac{u^{19}}{19}

          Now substitute uu back in:

          sin⁡19(x)19\frac{\sin^{19}{\left(x \right)}}{19}

        So, the result is: −715sin⁡19(x)19- \frac{715 \sin^{19}{\left(x \right)}}{19}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫1287sin⁡16(x)cos⁡(x) dx=1287∫sin⁡16(x)cos⁡(x) dx\int 1287 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx = 1287 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u16 du\int u^{16}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u16 du=u1717\int u^{16}\, du = \frac{u^{17}}{17}

          Now substitute uu back in:

          sin⁡17(x)17\frac{\sin^{17}{\left(x \right)}}{17}

        So, the result is: 1287sin⁡17(x)17\frac{1287 \sin^{17}{\left(x \right)}}{17}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−1716sin⁡14(x)cos⁡(x)) dx=−1716∫sin⁡14(x)cos⁡(x) dx\int \left(- 1716 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1716 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u14 du\int u^{14}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u14 du=u1515\int u^{14}\, du = \frac{u^{15}}{15}

          Now substitute uu back in:

          sin⁡15(x)15\frac{\sin^{15}{\left(x \right)}}{15}

        So, the result is: −572sin⁡15(x)5- \frac{572 \sin^{15}{\left(x \right)}}{5}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫1716sin⁡12(x)cos⁡(x) dx=1716∫sin⁡12(x)cos⁡(x) dx\int 1716 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx = 1716 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u12 du\int u^{12}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u12 du=u1313\int u^{12}\, du = \frac{u^{13}}{13}

          Now substitute uu back in:

          sin⁡13(x)13\frac{\sin^{13}{\left(x \right)}}{13}

        So, the result is: 132sin⁡13(x)132 \sin^{13}{\left(x \right)}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−1287sin⁡10(x)cos⁡(x)) dx=−1287∫sin⁡10(x)cos⁡(x) dx\int \left(- 1287 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1287 \int \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u10 du\int u^{10}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u10 du=u1111\int u^{10}\, du = \frac{u^{11}}{11}

          Now substitute uu back in:

          sin⁡11(x)11\frac{\sin^{11}{\left(x \right)}}{11}

        So, the result is: −117sin⁡11(x)- 117 \sin^{11}{\left(x \right)}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫715sin⁡8(x)cos⁡(x) dx=715∫sin⁡8(x)cos⁡(x) dx\int 715 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx = 715 \int \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u8 du\int u^{8}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u8 du=u99\int u^{8}\, du = \frac{u^{9}}{9}

          Now substitute uu back in:

          sin⁡9(x)9\frac{\sin^{9}{\left(x \right)}}{9}

        So, the result is: 715sin⁡9(x)9\frac{715 \sin^{9}{\left(x \right)}}{9}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−286sin⁡6(x)cos⁡(x)) dx=−286∫sin⁡6(x)cos⁡(x) dx\int \left(- 286 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 286 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u6 du\int u^{6}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u6 du=u77\int u^{6}\, du = \frac{u^{7}}{7}

          Now substitute uu back in:

          sin⁡7(x)7\frac{\sin^{7}{\left(x \right)}}{7}

        So, the result is: −286sin⁡7(x)7- \frac{286 \sin^{7}{\left(x \right)}}{7}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫78sin⁡4(x)cos⁡(x) dx=78∫sin⁡4(x)cos⁡(x) dx\int 78 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 78 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u4 du\int u^{4}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

          Now substitute uu back in:

          sin⁡5(x)5\frac{\sin^{5}{\left(x \right)}}{5}

        So, the result is: 78sin⁡5(x)5\frac{78 \sin^{5}{\left(x \right)}}{5}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−13sin⁡2(x)cos⁡(x)) dx=−13∫sin⁡2(x)cos⁡(x) dx\int \left(- 13 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 13 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u2 du\int u^{2}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

          Now substitute uu back in:

          sin⁡3(x)3\frac{\sin^{3}{\left(x \right)}}{3}

        So, the result is: −13sin⁡3(x)3- \frac{13 \sin^{3}{\left(x \right)}}{3}

      1. The integral of cosine is sine:

        ∫cos⁡(x) dx=sin⁡(x)\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}

      The result is: −sin⁡27(x)27+13sin⁡25(x)25−78sin⁡23(x)23+286sin⁡21(x)21−715sin⁡19(x)19+1287sin⁡17(x)17−572sin⁡15(x)5+132sin⁡13(x)−117sin⁡11(x)+715sin⁡9(x)9−286sin⁡7(x)7+78sin⁡5(x)5−13sin⁡3(x)3+sin⁡(x)- \frac{\sin^{27}{\left(x \right)}}{27} + \frac{13 \sin^{25}{\left(x \right)}}{25} - \frac{78 \sin^{23}{\left(x \right)}}{23} + \frac{286 \sin^{21}{\left(x \right)}}{21} - \frac{715 \sin^{19}{\left(x \right)}}{19} + \frac{1287 \sin^{17}{\left(x \right)}}{17} - \frac{572 \sin^{15}{\left(x \right)}}{5} + 132 \sin^{13}{\left(x \right)} - 117 \sin^{11}{\left(x \right)} + \frac{715 \sin^{9}{\left(x \right)}}{9} - \frac{286 \sin^{7}{\left(x \right)}}{7} + \frac{78 \sin^{5}{\left(x \right)}}{5} - \frac{13 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}

    Method #2

    1. Rewrite the integrand:

      (1−sin⁡2(x))13cos⁡(x)=−sin⁡26(x)cos⁡(x)+13sin⁡24(x)cos⁡(x)−78sin⁡22(x)cos⁡(x)+286sin⁡20(x)cos⁡(x)−715sin⁡18(x)cos⁡(x)+1287sin⁡16(x)cos⁡(x)−1716sin⁡14(x)cos⁡(x)+1716sin⁡12(x)cos⁡(x)−1287sin⁡10(x)cos⁡(x)+715sin⁡8(x)cos⁡(x)−286sin⁡6(x)cos⁡(x)+78sin⁡4(x)cos⁡(x)−13sin⁡2(x)cos⁡(x)+cos⁡(x)\left(1 - \sin^{2}{\left(x \right)}\right)^{13} \cos{\left(x \right)} = - \sin^{26}{\left(x \right)} \cos{\left(x \right)} + 13 \sin^{24}{\left(x \right)} \cos{\left(x \right)} - 78 \sin^{22}{\left(x \right)} \cos{\left(x \right)} + 286 \sin^{20}{\left(x \right)} \cos{\left(x \right)} - 715 \sin^{18}{\left(x \right)} \cos{\left(x \right)} + 1287 \sin^{16}{\left(x \right)} \cos{\left(x \right)} - 1716 \sin^{14}{\left(x \right)} \cos{\left(x \right)} + 1716 \sin^{12}{\left(x \right)} \cos{\left(x \right)} - 1287 \sin^{10}{\left(x \right)} \cos{\left(x \right)} + 715 \sin^{8}{\left(x \right)} \cos{\left(x \right)} - 286 \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 78 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 13 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}

    2. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−sin⁡26(x)cos⁡(x)) dx=−∫sin⁡26(x)cos⁡(x) dx\int \left(- \sin^{26}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{26}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u26 du\int u^{26}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u26 du=u2727\int u^{26}\, du = \frac{u^{27}}{27}

          Now substitute uu back in:

          sin⁡27(x)27\frac{\sin^{27}{\left(x \right)}}{27}

        So, the result is: −sin⁡27(x)27- \frac{\sin^{27}{\left(x \right)}}{27}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫13sin⁡24(x)cos⁡(x) dx=13∫sin⁡24(x)cos⁡(x) dx\int 13 \sin^{24}{\left(x \right)} \cos{\left(x \right)}\, dx = 13 \int \sin^{24}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u24 du\int u^{24}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u24 du=u2525\int u^{24}\, du = \frac{u^{25}}{25}

          Now substitute uu back in:

          sin⁡25(x)25\frac{\sin^{25}{\left(x \right)}}{25}

        So, the result is: 13sin⁡25(x)25\frac{13 \sin^{25}{\left(x \right)}}{25}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−78sin⁡22(x)cos⁡(x)) dx=−78∫sin⁡22(x)cos⁡(x) dx\int \left(- 78 \sin^{22}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 78 \int \sin^{22}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u22 du\int u^{22}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u22 du=u2323\int u^{22}\, du = \frac{u^{23}}{23}

          Now substitute uu back in:

          sin⁡23(x)23\frac{\sin^{23}{\left(x \right)}}{23}

        So, the result is: −78sin⁡23(x)23- \frac{78 \sin^{23}{\left(x \right)}}{23}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫286sin⁡20(x)cos⁡(x) dx=286∫sin⁡20(x)cos⁡(x) dx\int 286 \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx = 286 \int \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u20 du\int u^{20}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u20 du=u2121\int u^{20}\, du = \frac{u^{21}}{21}

          Now substitute uu back in:

          sin⁡21(x)21\frac{\sin^{21}{\left(x \right)}}{21}

        So, the result is: 286sin⁡21(x)21\frac{286 \sin^{21}{\left(x \right)}}{21}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−715sin⁡18(x)cos⁡(x)) dx=−715∫sin⁡18(x)cos⁡(x) dx\int \left(- 715 \sin^{18}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 715 \int \sin^{18}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u18 du\int u^{18}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u18 du=u1919\int u^{18}\, du = \frac{u^{19}}{19}

          Now substitute uu back in:

          sin⁡19(x)19\frac{\sin^{19}{\left(x \right)}}{19}

        So, the result is: −715sin⁡19(x)19- \frac{715 \sin^{19}{\left(x \right)}}{19}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫1287sin⁡16(x)cos⁡(x) dx=1287∫sin⁡16(x)cos⁡(x) dx\int 1287 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx = 1287 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u16 du\int u^{16}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u16 du=u1717\int u^{16}\, du = \frac{u^{17}}{17}

          Now substitute uu back in:

          sin⁡17(x)17\frac{\sin^{17}{\left(x \right)}}{17}

        So, the result is: 1287sin⁡17(x)17\frac{1287 \sin^{17}{\left(x \right)}}{17}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−1716sin⁡14(x)cos⁡(x)) dx=−1716∫sin⁡14(x)cos⁡(x) dx\int \left(- 1716 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1716 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u14 du\int u^{14}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u14 du=u1515\int u^{14}\, du = \frac{u^{15}}{15}

          Now substitute uu back in:

          sin⁡15(x)15\frac{\sin^{15}{\left(x \right)}}{15}

        So, the result is: −572sin⁡15(x)5- \frac{572 \sin^{15}{\left(x \right)}}{5}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫1716sin⁡12(x)cos⁡(x) dx=1716∫sin⁡12(x)cos⁡(x) dx\int 1716 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx = 1716 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u12 du\int u^{12}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u12 du=u1313\int u^{12}\, du = \frac{u^{13}}{13}

          Now substitute uu back in:

          sin⁡13(x)13\frac{\sin^{13}{\left(x \right)}}{13}

        So, the result is: 132sin⁡13(x)132 \sin^{13}{\left(x \right)}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−1287sin⁡10(x)cos⁡(x)) dx=−1287∫sin⁡10(x)cos⁡(x) dx\int \left(- 1287 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1287 \int \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u10 du\int u^{10}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u10 du=u1111\int u^{10}\, du = \frac{u^{11}}{11}

          Now substitute uu back in:

          sin⁡11(x)11\frac{\sin^{11}{\left(x \right)}}{11}

        So, the result is: −117sin⁡11(x)- 117 \sin^{11}{\left(x \right)}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫715sin⁡8(x)cos⁡(x) dx=715∫sin⁡8(x)cos⁡(x) dx\int 715 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx = 715 \int \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u8 du\int u^{8}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u8 du=u99\int u^{8}\, du = \frac{u^{9}}{9}

          Now substitute uu back in:

          sin⁡9(x)9\frac{\sin^{9}{\left(x \right)}}{9}

        So, the result is: 715sin⁡9(x)9\frac{715 \sin^{9}{\left(x \right)}}{9}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−286sin⁡6(x)cos⁡(x)) dx=−286∫sin⁡6(x)cos⁡(x) dx\int \left(- 286 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 286 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u6 du\int u^{6}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u6 du=u77\int u^{6}\, du = \frac{u^{7}}{7}

          Now substitute uu back in:

          sin⁡7(x)7\frac{\sin^{7}{\left(x \right)}}{7}

        So, the result is: −286sin⁡7(x)7- \frac{286 \sin^{7}{\left(x \right)}}{7}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫78sin⁡4(x)cos⁡(x) dx=78∫sin⁡4(x)cos⁡(x) dx\int 78 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 78 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u4 du\int u^{4}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

          Now substitute uu back in:

          sin⁡5(x)5\frac{\sin^{5}{\left(x \right)}}{5}

        So, the result is: 78sin⁡5(x)5\frac{78 \sin^{5}{\left(x \right)}}{5}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−13sin⁡2(x)cos⁡(x)) dx=−13∫sin⁡2(x)cos⁡(x) dx\int \left(- 13 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 13 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx

        1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

          Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          ∫u2 du\int u^{2}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

          Now substitute uu back in:

          sin⁡3(x)3\frac{\sin^{3}{\left(x \right)}}{3}

        So, the result is: −13sin⁡3(x)3- \frac{13 \sin^{3}{\left(x \right)}}{3}

      1. The integral of cosine is sine:

        ∫cos⁡(x) dx=sin⁡(x)\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}

      The result is: −sin⁡27(x)27+13sin⁡25(x)25−78sin⁡23(x)23+286sin⁡21(x)21−715sin⁡19(x)19+1287sin⁡17(x)17−572sin⁡15(x)5+132sin⁡13(x)−117sin⁡11(x)+715sin⁡9(x)9−286sin⁡7(x)7+78sin⁡5(x)5−13sin⁡3(x)3+sin⁡(x)- \frac{\sin^{27}{\left(x \right)}}{27} + \frac{13 \sin^{25}{\left(x \right)}}{25} - \frac{78 \sin^{23}{\left(x \right)}}{23} + \frac{286 \sin^{21}{\left(x \right)}}{21} - \frac{715 \sin^{19}{\left(x \right)}}{19} + \frac{1287 \sin^{17}{\left(x \right)}}{17} - \frac{572 \sin^{15}{\left(x \right)}}{5} + 132 \sin^{13}{\left(x \right)} - 117 \sin^{11}{\left(x \right)} + \frac{715 \sin^{9}{\left(x \right)}}{9} - \frac{286 \sin^{7}{\left(x \right)}}{7} + \frac{78 \sin^{5}{\left(x \right)}}{5} - \frac{13 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}

  3. Now simplify:

    (−1300075sin⁡26(x)+18253053sin⁡24(x)−119041650sin⁡22(x)+478056150sin⁡20(x)−1320944625sin⁡18(x)+2657429775sin⁡16(x)−4015671660sin⁡14(x)+4633467300sin⁡12(x)−4106936925sin⁡10(x)+2788660875sin⁡8(x)−1434168450sin⁡6(x)+547591590sin⁡4(x)−152108775sin⁡2(x)+35102025)sin⁡(x)35102025\frac{\left(- 1300075 \sin^{26}{\left(x \right)} + 18253053 \sin^{24}{\left(x \right)} - 119041650 \sin^{22}{\left(x \right)} + 478056150 \sin^{20}{\left(x \right)} - 1320944625 \sin^{18}{\left(x \right)} + 2657429775 \sin^{16}{\left(x \right)} - 4015671660 \sin^{14}{\left(x \right)} + 4633467300 \sin^{12}{\left(x \right)} - 4106936925 \sin^{10}{\left(x \right)} + 2788660875 \sin^{8}{\left(x \right)} - 1434168450 \sin^{6}{\left(x \right)} + 547591590 \sin^{4}{\left(x \right)} - 152108775 \sin^{2}{\left(x \right)} + 35102025\right) \sin{\left(x \right)}}{35102025}

  4. Add the constant of integration:

    (−1300075sin⁡26(x)+18253053sin⁡24(x)−119041650sin⁡22(x)+478056150sin⁡20(x)−1320944625sin⁡18(x)+2657429775sin⁡16(x)−4015671660sin⁡14(x)+4633467300sin⁡12(x)−4106936925sin⁡10(x)+2788660875sin⁡8(x)−1434168450sin⁡6(x)+547591590sin⁡4(x)−152108775sin⁡2(x)+35102025)sin⁡(x)35102025+constant\frac{\left(- 1300075 \sin^{26}{\left(x \right)} + 18253053 \sin^{24}{\left(x \right)} - 119041650 \sin^{22}{\left(x \right)} + 478056150 \sin^{20}{\left(x \right)} - 1320944625 \sin^{18}{\left(x \right)} + 2657429775 \sin^{16}{\left(x \right)} - 4015671660 \sin^{14}{\left(x \right)} + 4633467300 \sin^{12}{\left(x \right)} - 4106936925 \sin^{10}{\left(x \right)} + 2788660875 \sin^{8}{\left(x \right)} - 1434168450 \sin^{6}{\left(x \right)} + 547591590 \sin^{4}{\left(x \right)} - 152108775 \sin^{2}{\left(x \right)} + 35102025\right) \sin{\left(x \right)}}{35102025}+ \mathrm{constant}

The answer is:

(−1300075sin⁡26(x)+18253053sin⁡24(x)−119041650sin⁡22(x)+478056150sin⁡20(x)−1320944625sin⁡18(x)+2657429775sin⁡16(x)−4015671660sin⁡14(x)+4633467300sin⁡12(x)−4106936925sin⁡10(x)+2788660875sin⁡8(x)−1434168450sin⁡6(x)+547591590sin⁡4(x)−152108775sin⁡2(x)+35102025)sin⁡(x)35102025+constant\frac{\left(- 1300075 \sin^{26}{\left(x \right)} + 18253053 \sin^{24}{\left(x \right)} - 119041650 \sin^{22}{\left(x \right)} + 478056150 \sin^{20}{\left(x \right)} - 1320944625 \sin^{18}{\left(x \right)} + 2657429775 \sin^{16}{\left(x \right)} - 4015671660 \sin^{14}{\left(x \right)} + 4633467300 \sin^{12}{\left(x \right)} - 4106936925 \sin^{10}{\left(x \right)} + 2788660875 \sin^{8}{\left(x \right)} - 1434168450 \sin^{6}{\left(x \right)} + 547591590 \sin^{4}{\left(x \right)} - 152108775 \sin^{2}{\left(x \right)} + 35102025\right) \sin{\left(x \right)}}{35102025}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                                                                                                                                                                
 |                                                        19             15             7            23            3         27            25            5             21             9              17            
 |    27                    11             13      715*sin  (x)   572*sin  (x)   286*sin (x)   78*sin  (x)   13*sin (x)   sin  (x)   13*sin  (x)   78*sin (x)   286*sin  (x)   715*sin (x)   1287*sin  (x)         
 | cos  (x) dx = C - 117*sin  (x) + 132*sin  (x) - ------------ - ------------ - ----------- - ----------- - ---------- - -------- + ----------- + ---------- + ------------ + ----------- + ------------- + sin(x)
 |                                                      19             5              7             23           3           27           25           5             21             9              17              
/
∫cos⁡27(x) dx=C−sin⁡27(x)27+13sin⁡25(x)25−78sin⁡23(x)23+286sin⁡21(x)21−715sin⁡19(x)19+1287sin⁡17(x)17−572sin⁡15(x)5+132sin⁡13(x)−117sin⁡11(x)+715sin⁡9(x)9−286sin⁡7(x)7+78sin⁡5(x)5−13sin⁡3(x)3+sin⁡(x)\int \cos^{27}{\left(x \right)}\, dx = C - \frac{\sin^{27}{\left(x \right)}}{27} + \frac{13 \sin^{25}{\left(x \right)}}{25} - \frac{78 \sin^{23}{\left(x \right)}}{23} + \frac{286 \sin^{21}{\left(x \right)}}{21} - \frac{715 \sin^{19}{\left(x \right)}}{19} + \frac{1287 \sin^{17}{\left(x \right)}}{17} - \frac{572 \sin^{15}{\left(x \right)}}{5} + 132 \sin^{13}{\left(x \right)} - 117 \sin^{11}{\left(x \right)} + \frac{715 \sin^{9}{\left(x \right)}}{9} - \frac{286 \sin^{7}{\left(x \right)}}{7} + \frac{78 \sin^{5}{\left(x \right)}}{5} - \frac{13 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}
Simplify
The answer [src] 
                                                                17             9             21            5            25         27            3            23             7             15             19                                                                  17             9             21            5            25         27            3            23             7             15             19    
   22308732928                   13             11      1287*sin  (2)   715*sin (2)   286*sin  (2)   78*sin (2)   13*sin  (2)   sin  (2)   13*sin (2)   78*sin  (2)   286*sin (2)   572*sin  (2)   715*sin  (2)  22308732928                   13             11      1287*sin  (2)   715*sin (2)   286*sin  (2)   78*sin (2)   13*sin  (2)   sin  (2)   13*sin (2)   78*sin  (2)   286*sin (2)   572*sin  (2)   715*sin  (2) 
<- ----------- - sin(2) - 132*sin  (2) + 117*sin  (2) - ------------- - ----------- - ------------ - ---------- - ----------- + -------- + ---------- + ----------- + ----------- + ------------ + ------------, ----------- - sin(2) - 132*sin  (2) + 117*sin  (2) - ------------- - ----------- - ------------ - ---------- - ----------- + -------- + ---------- + ----------- + ----------- + ------------ + ------------>
     35102025                                                 17             9             21            5             25          27          3             23            7             5              19         35102025                                                 17             9             21            5             25          27          3             23            7             5              19
⟨−2230873292835102025−132sin⁡13(2)−715sin⁡9(2)9−1287sin⁡17(2)17−78sin⁡5(2)5−286sin⁡21(2)21−sin⁡(2)−13sin⁡25(2)25+sin⁡27(2)27+78sin⁡23(2)23+13sin⁡3(2)3+715sin⁡19(2)19+286sin⁡7(2)7+572sin⁡15(2)5+117sin⁡11(2),−132sin⁡13(2)−715sin⁡9(2)9−1287sin⁡17(2)17−78sin⁡5(2)5−286sin⁡21(2)21−sin⁡(2)−13sin⁡25(2)25+sin⁡27(2)27+78sin⁡23(2)23+13sin⁡3(2)3+715sin⁡19(2)19+286sin⁡7(2)7+572sin⁡15(2)5+117sin⁡11(2)+2230873292835102025⟩\left\langle - \frac{22308732928}{35102025} - 132 \sin^{13}{\left(2 \right)} - \frac{715 \sin^{9}{\left(2 \right)}}{9} - \frac{1287 \sin^{17}{\left(2 \right)}}{17} - \frac{78 \sin^{5}{\left(2 \right)}}{5} - \frac{286 \sin^{21}{\left(2 \right)}}{21} - \sin{\left(2 \right)} - \frac{13 \sin^{25}{\left(2 \right)}}{25} + \frac{\sin^{27}{\left(2 \right)}}{27} + \frac{78 \sin^{23}{\left(2 \right)}}{23} + \frac{13 \sin^{3}{\left(2 \right)}}{3} + \frac{715 \sin^{19}{\left(2 \right)}}{19} + \frac{286 \sin^{7}{\left(2 \right)}}{7} + \frac{572 \sin^{15}{\left(2 \right)}}{5} + 117 \sin^{11}{\left(2 \right)}, - 132 \sin^{13}{\left(2 \right)} - \frac{715 \sin^{9}{\left(2 \right)}}{9} - \frac{1287 \sin^{17}{\left(2 \right)}}{17} - \frac{78 \sin^{5}{\left(2 \right)}}{5} - \frac{286 \sin^{21}{\left(2 \right)}}{21} - \sin{\left(2 \right)} - \frac{13 \sin^{25}{\left(2 \right)}}{25} + \frac{\sin^{27}{\left(2 \right)}}{27} + \frac{78 \sin^{23}{\left(2 \right)}}{23} + \frac{13 \sin^{3}{\left(2 \right)}}{3} + \frac{715 \sin^{19}{\left(2 \right)}}{19} + \frac{286 \sin^{7}{\left(2 \right)}}{7} + \frac{572 \sin^{15}{\left(2 \right)}}{5} + 117 \sin^{11}{\left(2 \right)} + \frac{22308732928}{35102025}\right\rangle 
=
= 
                                                                17             9             21            5            25         27            3            23             7             15             19                                                                  17             9             21            5            25         27            3            23             7             15             19    
   22308732928                   13             11      1287*sin  (2)   715*sin (2)   286*sin  (2)   78*sin (2)   13*sin  (2)   sin  (2)   13*sin (2)   78*sin  (2)   286*sin (2)   572*sin  (2)   715*sin  (2)  22308732928                   13             11      1287*sin  (2)   715*sin (2)   286*sin  (2)   78*sin (2)   13*sin  (2)   sin  (2)   13*sin (2)   78*sin  (2)   286*sin (2)   572*sin  (2)   715*sin  (2) 
<- ----------- - sin(2) - 132*sin  (2) + 117*sin  (2) - ------------- - ----------- - ------------ - ---------- - ----------- + -------- + ---------- + ----------- + ----------- + ------------ + ------------, ----------- - sin(2) - 132*sin  (2) + 117*sin  (2) - ------------- - ----------- - ------------ - ---------- - ----------- + -------- + ---------- + ----------- + ----------- + ------------ + ------------>
     35102025                                                 17             9             21            5             25          27          3             23            7             5              19         35102025                                                 17             9             21            5             25          27          3             23            7             5              19
⟨−2230873292835102025−132sin⁡13(2)−715sin⁡9(2)9−1287sin⁡17(2)17−78sin⁡5(2)5−286sin⁡21(2)21−sin⁡(2)−13sin⁡25(2)25+sin⁡27(2)27+78sin⁡23(2)23+13sin⁡3(2)3+715sin⁡19(2)19+286sin⁡7(2)7+572sin⁡15(2)5+117sin⁡11(2),−132sin⁡13(2)−715sin⁡9(2)9−1287sin⁡17(2)17−78sin⁡5(2)5−286sin⁡21(2)21−sin⁡(2)−13sin⁡25(2)25+sin⁡27(2)27+78sin⁡23(2)23+13sin⁡3(2)3+715sin⁡19(2)19+286sin⁡7(2)7+572sin⁡15(2)5+117sin⁡11(2)+2230873292835102025⟩\left\langle - \frac{22308732928}{35102025} - 132 \sin^{13}{\left(2 \right)} - \frac{715 \sin^{9}{\left(2 \right)}}{9} - \frac{1287 \sin^{17}{\left(2 \right)}}{17} - \frac{78 \sin^{5}{\left(2 \right)}}{5} - \frac{286 \sin^{21}{\left(2 \right)}}{21} - \sin{\left(2 \right)} - \frac{13 \sin^{25}{\left(2 \right)}}{25} + \frac{\sin^{27}{\left(2 \right)}}{27} + \frac{78 \sin^{23}{\left(2 \right)}}{23} + \frac{13 \sin^{3}{\left(2 \right)}}{3} + \frac{715 \sin^{19}{\left(2 \right)}}{19} + \frac{286 \sin^{7}{\left(2 \right)}}{7} + \frac{572 \sin^{15}{\left(2 \right)}}{5} + 117 \sin^{11}{\left(2 \right)}, - 132 \sin^{13}{\left(2 \right)} - \frac{715 \sin^{9}{\left(2 \right)}}{9} - \frac{1287 \sin^{17}{\left(2 \right)}}{17} - \frac{78 \sin^{5}{\left(2 \right)}}{5} - \frac{286 \sin^{21}{\left(2 \right)}}{21} - \sin{\left(2 \right)} - \frac{13 \sin^{25}{\left(2 \right)}}{25} + \frac{\sin^{27}{\left(2 \right)}}{27} + \frac{78 \sin^{23}{\left(2 \right)}}{23} + \frac{13 \sin^{3}{\left(2 \right)}}{3} + \frac{715 \sin^{19}{\left(2 \right)}}{19} + \frac{286 \sin^{7}{\left(2 \right)}}{7} + \frac{572 \sin^{15}{\left(2 \right)}}{5} + 117 \sin^{11}{\left(2 \right)} + \frac{22308732928}{35102025}\right\rangle 
AccumBounds(-22308732928/35102025 - sin(2) - 132*sin(2)^13 + 117*sin(2)^11 - 1287*sin(2)^17/17 - 715*sin(2)^9/9 - 286*sin(2)^21/21 - 78*sin(2)^5/5 - 13*sin(2)^25/25 + sin(2)^27/27 + 13*sin(2)^3/3 + 78*sin(2)^23/23 + 286*sin(2)^7/7 + 572*sin(2)^15/5 + 715*sin(2)^19/19, 22308732928/35102025 - sin(2) - 132*sin(2)^13 + 117*sin(2)^11 - 1287*sin(2)^17/17 - 715*sin(2)^9/9 - 286*sin(2)^21/21 - 78*sin(2)^5/5 - 13*sin(2)^25/25 + sin(2)^27/27 + 13*sin(2)^3/3 + 78*sin(2)^23/23 + 286*sin(2)^7/7 + 572*sin(2)^15/5 + 715*sin(2)^19/19)

    Use the examples entering the upper and lower limits of integration.

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