Mister Exam

Other calculators

sin²5x
Solving integrals/ sin²5x

Integral of sin²5x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1            
  /            
 |             
 |     25      
 |  sin  (x) dx
 |             
/              
0
∫01sin⁡25(x) dx\int\limits_{0}^{1} \sin^{25}{\left(x \right)}\, dx 
Integral(sin(x)^25, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    sin⁡25(x)=(1−cos⁡2(x))12sin⁡(x)\sin^{25}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{12} \sin{\left(x \right)}

  2. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      (1−cos⁡2(x))12sin⁡(x)=sin⁡(x)cos⁡24(x)−12sin⁡(x)cos⁡22(x)+66sin⁡(x)cos⁡20(x)−220sin⁡(x)cos⁡18(x)+495sin⁡(x)cos⁡16(x)−792sin⁡(x)cos⁡14(x)+924sin⁡(x)cos⁡12(x)−792sin⁡(x)cos⁡10(x)+495sin⁡(x)cos⁡8(x)−220sin⁡(x)cos⁡6(x)+66sin⁡(x)cos⁡4(x)−12sin⁡(x)cos⁡2(x)+sin⁡(x)\left(1 - \cos^{2}{\left(x \right)}\right)^{12} \sin{\left(x \right)} = \sin{\left(x \right)} \cos^{24}{\left(x \right)} - 12 \sin{\left(x \right)} \cos^{22}{\left(x \right)} + 66 \sin{\left(x \right)} \cos^{20}{\left(x \right)} - 220 \sin{\left(x \right)} \cos^{18}{\left(x \right)} + 495 \sin{\left(x \right)} \cos^{16}{\left(x \right)} - 792 \sin{\left(x \right)} \cos^{14}{\left(x \right)} + 924 \sin{\left(x \right)} \cos^{12}{\left(x \right)} - 792 \sin{\left(x \right)} \cos^{10}{\left(x \right)} + 495 \sin{\left(x \right)} \cos^{8}{\left(x \right)} - 220 \sin{\left(x \right)} \cos^{6}{\left(x \right)} + 66 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(x \right)}

    2. Integrate term-by-term:

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

        Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

        ∫(−u24) du\int \left(- u^{24}\right)\, du

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

          ∫u24 du=−∫u24 du\int u^{24}\, 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}

          So, the result is: −u2525- \frac{u^{25}}{25}

        Now substitute uu back in:

        −cos⁡25(x)25- \frac{\cos^{25}{\left(x \right)}}{25}

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

        ∫(−12sin⁡(x)cos⁡22(x)) dx=−12∫sin⁡(x)cos⁡22(x) dx\int \left(- 12 \sin{\left(x \right)} \cos^{22}{\left(x \right)}\right)\, dx = - 12 \int \sin{\left(x \right)} \cos^{22}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u22) du\int \left(- u^{22}\right)\, du

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

            ∫u22 du=−∫u22 du\int u^{22}\, 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}

            So, the result is: −u2323- \frac{u^{23}}{23}

          Now substitute uu back in:

          −cos⁡23(x)23- \frac{\cos^{23}{\left(x \right)}}{23}

        So, the result is: 12cos⁡23(x)23\frac{12 \cos^{23}{\left(x \right)}}{23}

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

        ∫66sin⁡(x)cos⁡20(x) dx=66∫sin⁡(x)cos⁡20(x) dx\int 66 \sin{\left(x \right)} \cos^{20}{\left(x \right)}\, dx = 66 \int \sin{\left(x \right)} \cos^{20}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u20) du\int \left(- u^{20}\right)\, du

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

            ∫u20 du=−∫u20 du\int u^{20}\, 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}

            So, the result is: −u2121- \frac{u^{21}}{21}

          Now substitute uu back in:

          −cos⁡21(x)21- \frac{\cos^{21}{\left(x \right)}}{21}

        So, the result is: −22cos⁡21(x)7- \frac{22 \cos^{21}{\left(x \right)}}{7}

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

        ∫(−220sin⁡(x)cos⁡18(x)) dx=−220∫sin⁡(x)cos⁡18(x) dx\int \left(- 220 \sin{\left(x \right)} \cos^{18}{\left(x \right)}\right)\, dx = - 220 \int \sin{\left(x \right)} \cos^{18}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u18) du\int \left(- u^{18}\right)\, du

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

            ∫u18 du=−∫u18 du\int u^{18}\, 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}

            So, the result is: −u1919- \frac{u^{19}}{19}

          Now substitute uu back in:

          −cos⁡19(x)19- \frac{\cos^{19}{\left(x \right)}}{19}

        So, the result is: 220cos⁡19(x)19\frac{220 \cos^{19}{\left(x \right)}}{19}

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

        ∫495sin⁡(x)cos⁡16(x) dx=495∫sin⁡(x)cos⁡16(x) dx\int 495 \sin{\left(x \right)} \cos^{16}{\left(x \right)}\, dx = 495 \int \sin{\left(x \right)} \cos^{16}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u16) du\int \left(- u^{16}\right)\, du

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

            ∫u16 du=−∫u16 du\int u^{16}\, 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}

            So, the result is: −u1717- \frac{u^{17}}{17}

          Now substitute uu back in:

          −cos⁡17(x)17- \frac{\cos^{17}{\left(x \right)}}{17}

        So, the result is: −495cos⁡17(x)17- \frac{495 \cos^{17}{\left(x \right)}}{17}

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

        ∫(−792sin⁡(x)cos⁡14(x)) dx=−792∫sin⁡(x)cos⁡14(x) dx\int \left(- 792 \sin{\left(x \right)} \cos^{14}{\left(x \right)}\right)\, dx = - 792 \int \sin{\left(x \right)} \cos^{14}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u14) du\int \left(- u^{14}\right)\, du

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

            ∫u14 du=−∫u14 du\int u^{14}\, 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}

            So, the result is: −u1515- \frac{u^{15}}{15}

          Now substitute uu back in:

          −cos⁡15(x)15- \frac{\cos^{15}{\left(x \right)}}{15}

        So, the result is: 264cos⁡15(x)5\frac{264 \cos^{15}{\left(x \right)}}{5}

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

        ∫924sin⁡(x)cos⁡12(x) dx=924∫sin⁡(x)cos⁡12(x) dx\int 924 \sin{\left(x \right)} \cos^{12}{\left(x \right)}\, dx = 924 \int \sin{\left(x \right)} \cos^{12}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u12) du\int \left(- u^{12}\right)\, du

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

            ∫u12 du=−∫u12 du\int u^{12}\, 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}

            So, the result is: −u1313- \frac{u^{13}}{13}

          Now substitute uu back in:

          −cos⁡13(x)13- \frac{\cos^{13}{\left(x \right)}}{13}

        So, the result is: −924cos⁡13(x)13- \frac{924 \cos^{13}{\left(x \right)}}{13}

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

        ∫(−792sin⁡(x)cos⁡10(x)) dx=−792∫sin⁡(x)cos⁡10(x) dx\int \left(- 792 \sin{\left(x \right)} \cos^{10}{\left(x \right)}\right)\, dx = - 792 \int \sin{\left(x \right)} \cos^{10}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u10) du\int \left(- u^{10}\right)\, du

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

            ∫u10 du=−∫u10 du\int u^{10}\, 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}

            So, the result is: −u1111- \frac{u^{11}}{11}

          Now substitute uu back in:

          −cos⁡11(x)11- \frac{\cos^{11}{\left(x \right)}}{11}

        So, the result is: 72cos⁡11(x)72 \cos^{11}{\left(x \right)}

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

        ∫495sin⁡(x)cos⁡8(x) dx=495∫sin⁡(x)cos⁡8(x) dx\int 495 \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx = 495 \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u8) du\int \left(- u^{8}\right)\, du

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

            ∫u8 du=−∫u8 du\int u^{8}\, 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}

            So, the result is: −u99- \frac{u^{9}}{9}

          Now substitute uu back in:

          −cos⁡9(x)9- \frac{\cos^{9}{\left(x \right)}}{9}

        So, the result is: −55cos⁡9(x)- 55 \cos^{9}{\left(x \right)}

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

        ∫(−220sin⁡(x)cos⁡6(x)) dx=−220∫sin⁡(x)cos⁡6(x) dx\int \left(- 220 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - 220 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u6) du\int \left(- u^{6}\right)\, du

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

            ∫u6 du=−∫u6 du\int u^{6}\, 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}

            So, the result is: −u77- \frac{u^{7}}{7}

          Now substitute uu back in:

          −cos⁡7(x)7- \frac{\cos^{7}{\left(x \right)}}{7}

        So, the result is: 220cos⁡7(x)7\frac{220 \cos^{7}{\left(x \right)}}{7}

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

        ∫66sin⁡(x)cos⁡4(x) dx=66∫sin⁡(x)cos⁡4(x) dx\int 66 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 66 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u4) du\int \left(- u^{4}\right)\, du

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

            ∫u4 du=−∫u4 du\int u^{4}\, 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}

            So, the result is: −u55- \frac{u^{5}}{5}

          Now substitute uu back in:

          −cos⁡5(x)5- \frac{\cos^{5}{\left(x \right)}}{5}

        So, the result is: −66cos⁡5(x)5- \frac{66 \cos^{5}{\left(x \right)}}{5}

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

        ∫(−12sin⁡(x)cos⁡2(x)) dx=−12∫sin⁡(x)cos⁡2(x) dx\int \left(- 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 12 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u2) du\int \left(- u^{2}\right)\, du

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

            ∫u2 du=−∫u2 du\int u^{2}\, 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}

            So, the result is: −u33- \frac{u^{3}}{3}

          Now substitute uu back in:

          −cos⁡3(x)3- \frac{\cos^{3}{\left(x \right)}}{3}

        So, the result is: 4cos⁡3(x)4 \cos^{3}{\left(x \right)}

      1. The integral of sine is negative cosine:

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

      The result is: −cos⁡25(x)25+12cos⁡23(x)23−22cos⁡21(x)7+220cos⁡19(x)19−495cos⁡17(x)17+264cos⁡15(x)5−924cos⁡13(x)13+72cos⁡11(x)−55cos⁡9(x)+220cos⁡7(x)7−66cos⁡5(x)5+4cos⁡3(x)−cos⁡(x)- \frac{\cos^{25}{\left(x \right)}}{25} + \frac{12 \cos^{23}{\left(x \right)}}{23} - \frac{22 \cos^{21}{\left(x \right)}}{7} + \frac{220 \cos^{19}{\left(x \right)}}{19} - \frac{495 \cos^{17}{\left(x \right)}}{17} + \frac{264 \cos^{15}{\left(x \right)}}{5} - \frac{924 \cos^{13}{\left(x \right)}}{13} + 72 \cos^{11}{\left(x \right)} - 55 \cos^{9}{\left(x \right)} + \frac{220 \cos^{7}{\left(x \right)}}{7} - \frac{66 \cos^{5}{\left(x \right)}}{5} + 4 \cos^{3}{\left(x \right)} - \cos{\left(x \right)}

    Method #2

    1. Rewrite the integrand:

      (1−cos⁡2(x))12sin⁡(x)=sin⁡(x)cos⁡24(x)−12sin⁡(x)cos⁡22(x)+66sin⁡(x)cos⁡20(x)−220sin⁡(x)cos⁡18(x)+495sin⁡(x)cos⁡16(x)−792sin⁡(x)cos⁡14(x)+924sin⁡(x)cos⁡12(x)−792sin⁡(x)cos⁡10(x)+495sin⁡(x)cos⁡8(x)−220sin⁡(x)cos⁡6(x)+66sin⁡(x)cos⁡4(x)−12sin⁡(x)cos⁡2(x)+sin⁡(x)\left(1 - \cos^{2}{\left(x \right)}\right)^{12} \sin{\left(x \right)} = \sin{\left(x \right)} \cos^{24}{\left(x \right)} - 12 \sin{\left(x \right)} \cos^{22}{\left(x \right)} + 66 \sin{\left(x \right)} \cos^{20}{\left(x \right)} - 220 \sin{\left(x \right)} \cos^{18}{\left(x \right)} + 495 \sin{\left(x \right)} \cos^{16}{\left(x \right)} - 792 \sin{\left(x \right)} \cos^{14}{\left(x \right)} + 924 \sin{\left(x \right)} \cos^{12}{\left(x \right)} - 792 \sin{\left(x \right)} \cos^{10}{\left(x \right)} + 495 \sin{\left(x \right)} \cos^{8}{\left(x \right)} - 220 \sin{\left(x \right)} \cos^{6}{\left(x \right)} + 66 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(x \right)}

    2. Integrate term-by-term:

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

        Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

        ∫(−u24) du\int \left(- u^{24}\right)\, du

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

          ∫u24 du=−∫u24 du\int u^{24}\, 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}

          So, the result is: −u2525- \frac{u^{25}}{25}

        Now substitute uu back in:

        −cos⁡25(x)25- \frac{\cos^{25}{\left(x \right)}}{25}

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

        ∫(−12sin⁡(x)cos⁡22(x)) dx=−12∫sin⁡(x)cos⁡22(x) dx\int \left(- 12 \sin{\left(x \right)} \cos^{22}{\left(x \right)}\right)\, dx = - 12 \int \sin{\left(x \right)} \cos^{22}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u22) du\int \left(- u^{22}\right)\, du

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

            ∫u22 du=−∫u22 du\int u^{22}\, 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}

            So, the result is: −u2323- \frac{u^{23}}{23}

          Now substitute uu back in:

          −cos⁡23(x)23- \frac{\cos^{23}{\left(x \right)}}{23}

        So, the result is: 12cos⁡23(x)23\frac{12 \cos^{23}{\left(x \right)}}{23}

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

        ∫66sin⁡(x)cos⁡20(x) dx=66∫sin⁡(x)cos⁡20(x) dx\int 66 \sin{\left(x \right)} \cos^{20}{\left(x \right)}\, dx = 66 \int \sin{\left(x \right)} \cos^{20}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u20) du\int \left(- u^{20}\right)\, du

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

            ∫u20 du=−∫u20 du\int u^{20}\, 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}

            So, the result is: −u2121- \frac{u^{21}}{21}

          Now substitute uu back in:

          −cos⁡21(x)21- \frac{\cos^{21}{\left(x \right)}}{21}

        So, the result is: −22cos⁡21(x)7- \frac{22 \cos^{21}{\left(x \right)}}{7}

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

        ∫(−220sin⁡(x)cos⁡18(x)) dx=−220∫sin⁡(x)cos⁡18(x) dx\int \left(- 220 \sin{\left(x \right)} \cos^{18}{\left(x \right)}\right)\, dx = - 220 \int \sin{\left(x \right)} \cos^{18}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u18) du\int \left(- u^{18}\right)\, du

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

            ∫u18 du=−∫u18 du\int u^{18}\, 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}

            So, the result is: −u1919- \frac{u^{19}}{19}

          Now substitute uu back in:

          −cos⁡19(x)19- \frac{\cos^{19}{\left(x \right)}}{19}

        So, the result is: 220cos⁡19(x)19\frac{220 \cos^{19}{\left(x \right)}}{19}

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

        ∫495sin⁡(x)cos⁡16(x) dx=495∫sin⁡(x)cos⁡16(x) dx\int 495 \sin{\left(x \right)} \cos^{16}{\left(x \right)}\, dx = 495 \int \sin{\left(x \right)} \cos^{16}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u16) du\int \left(- u^{16}\right)\, du

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

            ∫u16 du=−∫u16 du\int u^{16}\, 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}

            So, the result is: −u1717- \frac{u^{17}}{17}

          Now substitute uu back in:

          −cos⁡17(x)17- \frac{\cos^{17}{\left(x \right)}}{17}

        So, the result is: −495cos⁡17(x)17- \frac{495 \cos^{17}{\left(x \right)}}{17}

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

        ∫(−792sin⁡(x)cos⁡14(x)) dx=−792∫sin⁡(x)cos⁡14(x) dx\int \left(- 792 \sin{\left(x \right)} \cos^{14}{\left(x \right)}\right)\, dx = - 792 \int \sin{\left(x \right)} \cos^{14}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u14) du\int \left(- u^{14}\right)\, du

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

            ∫u14 du=−∫u14 du\int u^{14}\, 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}

            So, the result is: −u1515- \frac{u^{15}}{15}

          Now substitute uu back in:

          −cos⁡15(x)15- \frac{\cos^{15}{\left(x \right)}}{15}

        So, the result is: 264cos⁡15(x)5\frac{264 \cos^{15}{\left(x \right)}}{5}

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

        ∫924sin⁡(x)cos⁡12(x) dx=924∫sin⁡(x)cos⁡12(x) dx\int 924 \sin{\left(x \right)} \cos^{12}{\left(x \right)}\, dx = 924 \int \sin{\left(x \right)} \cos^{12}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u12) du\int \left(- u^{12}\right)\, du

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

            ∫u12 du=−∫u12 du\int u^{12}\, 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}

            So, the result is: −u1313- \frac{u^{13}}{13}

          Now substitute uu back in:

          −cos⁡13(x)13- \frac{\cos^{13}{\left(x \right)}}{13}

        So, the result is: −924cos⁡13(x)13- \frac{924 \cos^{13}{\left(x \right)}}{13}

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

        ∫(−792sin⁡(x)cos⁡10(x)) dx=−792∫sin⁡(x)cos⁡10(x) dx\int \left(- 792 \sin{\left(x \right)} \cos^{10}{\left(x \right)}\right)\, dx = - 792 \int \sin{\left(x \right)} \cos^{10}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u10) du\int \left(- u^{10}\right)\, du

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

            ∫u10 du=−∫u10 du\int u^{10}\, 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}

            So, the result is: −u1111- \frac{u^{11}}{11}

          Now substitute uu back in:

          −cos⁡11(x)11- \frac{\cos^{11}{\left(x \right)}}{11}

        So, the result is: 72cos⁡11(x)72 \cos^{11}{\left(x \right)}

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

        ∫495sin⁡(x)cos⁡8(x) dx=495∫sin⁡(x)cos⁡8(x) dx\int 495 \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx = 495 \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u8) du\int \left(- u^{8}\right)\, du

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

            ∫u8 du=−∫u8 du\int u^{8}\, 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}

            So, the result is: −u99- \frac{u^{9}}{9}

          Now substitute uu back in:

          −cos⁡9(x)9- \frac{\cos^{9}{\left(x \right)}}{9}

        So, the result is: −55cos⁡9(x)- 55 \cos^{9}{\left(x \right)}

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

        ∫(−220sin⁡(x)cos⁡6(x)) dx=−220∫sin⁡(x)cos⁡6(x) dx\int \left(- 220 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - 220 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u6) du\int \left(- u^{6}\right)\, du

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

            ∫u6 du=−∫u6 du\int u^{6}\, 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}

            So, the result is: −u77- \frac{u^{7}}{7}

          Now substitute uu back in:

          −cos⁡7(x)7- \frac{\cos^{7}{\left(x \right)}}{7}

        So, the result is: 220cos⁡7(x)7\frac{220 \cos^{7}{\left(x \right)}}{7}

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

        ∫66sin⁡(x)cos⁡4(x) dx=66∫sin⁡(x)cos⁡4(x) dx\int 66 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 66 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u4) du\int \left(- u^{4}\right)\, du

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

            ∫u4 du=−∫u4 du\int u^{4}\, 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}

            So, the result is: −u55- \frac{u^{5}}{5}

          Now substitute uu back in:

          −cos⁡5(x)5- \frac{\cos^{5}{\left(x \right)}}{5}

        So, the result is: −66cos⁡5(x)5- \frac{66 \cos^{5}{\left(x \right)}}{5}

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

        ∫(−12sin⁡(x)cos⁡2(x)) dx=−12∫sin⁡(x)cos⁡2(x) dx\int \left(- 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 12 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫(−u2) du\int \left(- u^{2}\right)\, du

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

            ∫u2 du=−∫u2 du\int u^{2}\, 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}

            So, the result is: −u33- \frac{u^{3}}{3}

          Now substitute uu back in:

          −cos⁡3(x)3- \frac{\cos^{3}{\left(x \right)}}{3}

        So, the result is: 4cos⁡3(x)4 \cos^{3}{\left(x \right)}

      1. The integral of sine is negative cosine:

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

      The result is: −cos⁡25(x)25+12cos⁡23(x)23−22cos⁡21(x)7+220cos⁡19(x)19−495cos⁡17(x)17+264cos⁡15(x)5−924cos⁡13(x)13+72cos⁡11(x)−55cos⁡9(x)+220cos⁡7(x)7−66cos⁡5(x)5+4cos⁡3(x)−cos⁡(x)- \frac{\cos^{25}{\left(x \right)}}{25} + \frac{12 \cos^{23}{\left(x \right)}}{23} - \frac{22 \cos^{21}{\left(x \right)}}{7} + \frac{220 \cos^{19}{\left(x \right)}}{19} - \frac{495 \cos^{17}{\left(x \right)}}{17} + \frac{264 \cos^{15}{\left(x \right)}}{5} - \frac{924 \cos^{13}{\left(x \right)}}{13} + 72 \cos^{11}{\left(x \right)} - 55 \cos^{9}{\left(x \right)} + \frac{220 \cos^{7}{\left(x \right)}}{7} - \frac{66 \cos^{5}{\left(x \right)}}{5} + 4 \cos^{3}{\left(x \right)} - \cos{\left(x \right)}

  3. Now simplify:

    (−676039cos⁡24(x)+8817900cos⁡22(x)−53117350cos⁡20(x)+195695500cos⁡18(x)−492116625cos⁡16(x)+892371480cos⁡14(x)−1201269300cos⁡12(x)+1216870200cos⁡10(x)−929553625cos⁡8(x)+531173500cos⁡6(x)−223092870cos⁡4(x)+67603900cos⁡2(x)−16900975)cos⁡(x)16900975\frac{\left(- 676039 \cos^{24}{\left(x \right)} + 8817900 \cos^{22}{\left(x \right)} - 53117350 \cos^{20}{\left(x \right)} + 195695500 \cos^{18}{\left(x \right)} - 492116625 \cos^{16}{\left(x \right)} + 892371480 \cos^{14}{\left(x \right)} - 1201269300 \cos^{12}{\left(x \right)} + 1216870200 \cos^{10}{\left(x \right)} - 929553625 \cos^{8}{\left(x \right)} + 531173500 \cos^{6}{\left(x \right)} - 223092870 \cos^{4}{\left(x \right)} + 67603900 \cos^{2}{\left(x \right)} - 16900975\right) \cos{\left(x \right)}}{16900975}

  4. Add the constant of integration:

    (−676039cos⁡24(x)+8817900cos⁡22(x)−53117350cos⁡20(x)+195695500cos⁡18(x)−492116625cos⁡16(x)+892371480cos⁡14(x)−1201269300cos⁡12(x)+1216870200cos⁡10(x)−929553625cos⁡8(x)+531173500cos⁡6(x)−223092870cos⁡4(x)+67603900cos⁡2(x)−16900975)cos⁡(x)16900975+constant\frac{\left(- 676039 \cos^{24}{\left(x \right)} + 8817900 \cos^{22}{\left(x \right)} - 53117350 \cos^{20}{\left(x \right)} + 195695500 \cos^{18}{\left(x \right)} - 492116625 \cos^{16}{\left(x \right)} + 892371480 \cos^{14}{\left(x \right)} - 1201269300 \cos^{12}{\left(x \right)} + 1216870200 \cos^{10}{\left(x \right)} - 929553625 \cos^{8}{\left(x \right)} + 531173500 \cos^{6}{\left(x \right)} - 223092870 \cos^{4}{\left(x \right)} + 67603900 \cos^{2}{\left(x \right)} - 16900975\right) \cos{\left(x \right)}}{16900975}+ \mathrm{constant}

The answer is:

(−676039cos⁡24(x)+8817900cos⁡22(x)−53117350cos⁡20(x)+195695500cos⁡18(x)−492116625cos⁡16(x)+892371480cos⁡14(x)−1201269300cos⁡12(x)+1216870200cos⁡10(x)−929553625cos⁡8(x)+531173500cos⁡6(x)−223092870cos⁡4(x)+67603900cos⁡2(x)−16900975)cos⁡(x)16900975+constant\frac{\left(- 676039 \cos^{24}{\left(x \right)} + 8817900 \cos^{22}{\left(x \right)} - 53117350 \cos^{20}{\left(x \right)} + 195695500 \cos^{18}{\left(x \right)} - 492116625 \cos^{16}{\left(x \right)} + 892371480 \cos^{14}{\left(x \right)} - 1201269300 \cos^{12}{\left(x \right)} + 1216870200 \cos^{10}{\left(x \right)} - 929553625 \cos^{8}{\left(x \right)} + 531173500 \cos^{6}{\left(x \right)} - 223092870 \cos^{4}{\left(x \right)} + 67603900 \cos^{2}{\left(x \right)} - 16900975\right) \cos{\left(x \right)}}{16900975}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                                                                                                                                             
 |                                                                          13             17            5            21         25            23             7             19             15   
 |    25                            9           3            11      924*cos  (x)   495*cos  (x)   66*cos (x)   22*cos  (x)   cos  (x)   12*cos  (x)   220*cos (x)   220*cos  (x)   264*cos  (x)
 | sin  (x) dx = C - cos(x) - 55*cos (x) + 4*cos (x) + 72*cos  (x) - ------------ - ------------ - ---------- - ----------- - -------- + ----------- + ----------- + ------------ + ------------
 |                                                                        13             17            5             7           25           23            7             19             5      
/
∫sin⁡25(x) dx=C−cos⁡25(x)25+12cos⁡23(x)23−22cos⁡21(x)7+220cos⁡19(x)19−495cos⁡17(x)17+264cos⁡15(x)5−924cos⁡13(x)13+72cos⁡11(x)−55cos⁡9(x)+220cos⁡7(x)7−66cos⁡5(x)5+4cos⁡3(x)−cos⁡(x)\int \sin^{25}{\left(x \right)}\, dx = C - \frac{\cos^{25}{\left(x \right)}}{25} + \frac{12 \cos^{23}{\left(x \right)}}{23} - \frac{22 \cos^{21}{\left(x \right)}}{7} + \frac{220 \cos^{19}{\left(x \right)}}{19} - \frac{495 \cos^{17}{\left(x \right)}}{17} + \frac{264 \cos^{15}{\left(x \right)}}{5} - \frac{924 \cos^{13}{\left(x \right)}}{13} + 72 \cos^{11}{\left(x \right)} - 55 \cos^{9}{\left(x \right)} + \frac{220 \cos^{7}{\left(x \right)}}{7} - \frac{66 \cos^{5}{\left(x \right)}}{5} + 4 \cos^{3}{\left(x \right)} - \cos{\left(x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-0.500.25
Plot the graph f(x)
The answer [src] 
                                                                  13             17            5            21         25            23             7             19             15   
4194304                   9           3            11      924*cos  (1)   495*cos  (1)   66*cos (1)   22*cos  (1)   cos  (1)   12*cos  (1)   220*cos (1)   220*cos  (1)   264*cos  (1)
-------- - cos(1) - 55*cos (1) + 4*cos (1) + 72*cos  (1) - ------------ - ------------ - ---------- - ----------- - -------- + ----------- + ----------- + ------------ + ------------
16900975                                                        13             17            5             7           25           23            7             19             5
−66cos⁡5(1)5−cos⁡(1)−55cos⁡9(1)−924cos⁡13(1)13−495cos⁡17(1)17−22cos⁡21(1)7−cos⁡25(1)25+12cos⁡23(1)23+220cos⁡19(1)19+264cos⁡15(1)5+72cos⁡11(1)+419430416900975+220cos⁡7(1)7+4cos⁡3(1)- \frac{66 \cos^{5}{\left(1 \right)}}{5} - \cos{\left(1 \right)} - 55 \cos^{9}{\left(1 \right)} - \frac{924 \cos^{13}{\left(1 \right)}}{13} - \frac{495 \cos^{17}{\left(1 \right)}}{17} - \frac{22 \cos^{21}{\left(1 \right)}}{7} - \frac{\cos^{25}{\left(1 \right)}}{25} + \frac{12 \cos^{23}{\left(1 \right)}}{23} + \frac{220 \cos^{19}{\left(1 \right)}}{19} + \frac{264 \cos^{15}{\left(1 \right)}}{5} + 72 \cos^{11}{\left(1 \right)} + \frac{4194304}{16900975} + \frac{220 \cos^{7}{\left(1 \right)}}{7} + 4 \cos^{3}{\left(1 \right)} 
=
= 
                                                                  13             17            5            21         25            23             7             19             15   
4194304                   9           3            11      924*cos  (1)   495*cos  (1)   66*cos (1)   22*cos  (1)   cos  (1)   12*cos  (1)   220*cos (1)   220*cos  (1)   264*cos  (1)
-------- - cos(1) - 55*cos (1) + 4*cos (1) + 72*cos  (1) - ------------ - ------------ - ---------- - ----------- - -------- + ----------- + ----------- + ------------ + ------------
16900975                                                        13             17            5             7           25           23            7             19             5
−66cos⁡5(1)5−cos⁡(1)−55cos⁡9(1)−924cos⁡13(1)13−495cos⁡17(1)17−22cos⁡21(1)7−cos⁡25(1)25+12cos⁡23(1)23+220cos⁡19(1)19+264cos⁡15(1)5+72cos⁡11(1)+419430416900975+220cos⁡7(1)7+4cos⁡3(1)- \frac{66 \cos^{5}{\left(1 \right)}}{5} - \cos{\left(1 \right)} - 55 \cos^{9}{\left(1 \right)} - \frac{924 \cos^{13}{\left(1 \right)}}{13} - \frac{495 \cos^{17}{\left(1 \right)}}{17} - \frac{22 \cos^{21}{\left(1 \right)}}{7} - \frac{\cos^{25}{\left(1 \right)}}{25} + \frac{12 \cos^{23}{\left(1 \right)}}{23} + \frac{220 \cos^{19}{\left(1 \right)}}{19} + \frac{264 \cos^{15}{\left(1 \right)}}{5} + 72 \cos^{11}{\left(1 \right)} + \frac{4194304}{16900975} + \frac{220 \cos^{7}{\left(1 \right)}}{7} + 4 \cos^{3}{\left(1 \right)} 
4194304/16900975 - cos(1) - 55*cos(1)^9 + 4*cos(1)^3 + 72*cos(1)^11 - 924*cos(1)^13/13 - 495*cos(1)^17/17 - 66*cos(1)^5/5 - 22*cos(1)^21/7 - cos(1)^25/25 + 12*cos(1)^23/23 + 220*cos(1)^7/7 + 220*cos(1)^19/19 + 264*cos(1)^15/5
Numerical answer [src] 
0.000743706011186276
0.000743706011186276
The graph
Integral of sin²5x dx

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

    © Mister Exam – Calculator