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sin8x*cos3x
Solving integrals/ sin8x*cos3x

Integral of sin8x*cos3x dx

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∫01sin⁡(8x)cos⁡(3x) dx\int\limits_{0}^{1} \sin{\left(8 x \right)} \cos{\left(3 x \right)}\, dx
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Detail solution

  1. Rewrite the integrand:

    sin⁡(8x)cos⁡(3x)=−512sin⁡7(x)cos⁡4(x)+384sin⁡7(x)cos⁡2(x)+768sin⁡5(x)cos⁡4(x)−576sin⁡5(x)cos⁡2(x)−320sin⁡3(x)cos⁡4(x)+240sin⁡3(x)cos⁡2(x)+32sin⁡(x)cos⁡4(x)−24sin⁡(x)cos⁡2(x)\sin{\left(8 x \right)} \cos{\left(3 x \right)} = - 512 \sin^{7}{\left(x \right)} \cos^{4}{\left(x \right)} + 384 \sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)} + 768 \sin^{5}{\left(x \right)} \cos^{4}{\left(x \right)} - 576 \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)} - 320 \sin^{3}{\left(x \right)} \cos^{4}{\left(x \right)} + 240 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} + 32 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 24 \sin{\left(x \right)} \cos^{2}{\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:

      ∫(−512sin⁡7(x)cos⁡4(x)) dx=−512∫sin⁡7(x)cos⁡4(x) dx\int \left(- 512 \sin^{7}{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 512 \int \sin^{7}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx

      1. Rewrite the integrand:

        sin⁡7(x)cos⁡4(x)=(1−cos⁡2(x))3sin⁡(x)cos⁡4(x)\sin^{7}{\left(x \right)} \cos^{4}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{4}{\left(x \right)}

      2. There are multiple ways to do this integral.

        Method #1

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

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

          ∫(u10−3u8+3u6−u4) du\int \left(u^{10} - 3 u^{8} + 3 u^{6} - u^{4}\right)\, du

          1. Integrate term-by-term:

            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}

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

              ∫(−3u8) du=−3∫u8 du\int \left(- 3 u^{8}\right)\, du = - 3 \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: −u93- \frac{u^{9}}{3}

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

              ∫3u6 du=3∫u6 du\int 3 u^{6}\, du = 3 \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: 3u77\frac{3 u^{7}}{7}

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

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

            The result is: u1111−u93+3u77−u55\frac{u^{11}}{11} - \frac{u^{9}}{3} + \frac{3 u^{7}}{7} - \frac{u^{5}}{5}

          Now substitute uu back in:

          cos⁡11(x)11−cos⁡9(x)3+3cos⁡7(x)7−cos⁡5(x)5\frac{\cos^{11}{\left(x \right)}}{11} - \frac{\cos^{9}{\left(x \right)}}{3} + \frac{3 \cos^{7}{\left(x \right)}}{7} - \frac{\cos^{5}{\left(x \right)}}{5}

        Method #2

        1. Rewrite the integrand:

          (1−cos⁡2(x))3sin⁡(x)cos⁡4(x)=−sin⁡(x)cos⁡10(x)+3sin⁡(x)cos⁡8(x)−3sin⁡(x)cos⁡6(x)+sin⁡(x)cos⁡4(x)\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{4}{\left(x \right)} = - \sin{\left(x \right)} \cos^{10}{\left(x \right)} + 3 \sin{\left(x \right)} \cos^{8}{\left(x \right)} - 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)} + \sin{\left(x \right)} \cos^{4}{\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⁡(x)cos⁡10(x)) dx=−∫sin⁡(x)cos⁡10(x) dx\int \left(- \sin{\left(x \right)} \cos^{10}{\left(x \right)}\right)\, dx = - \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 u^{10}\, du

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

                ∫(−u10) du=−∫u10 du\int \left(- u^{10}\right)\, 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: cos⁡11(x)11\frac{\cos^{11}{\left(x \right)}}{11}

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

            ∫3sin⁡(x)cos⁡8(x) dx=3∫sin⁡(x)cos⁡8(x) dx\int 3 \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx = 3 \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 u^{8}\, du

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

                ∫(−u8) du=−∫u8 du\int \left(- u^{8}\right)\, 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: −cos⁡9(x)3- \frac{\cos^{9}{\left(x \right)}}{3}

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

            ∫(−3sin⁡(x)cos⁡6(x)) dx=−3∫sin⁡(x)cos⁡6(x) dx\int \left(- 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - 3 \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 u^{6}\, du

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

                ∫(−u6) du=−∫u6 du\int \left(- u^{6}\right)\, 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: 3cos⁡7(x)7\frac{3 \cos^{7}{\left(x \right)}}{7}

          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 u^{4}\, du

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

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

          The result is: cos⁡11(x)11−cos⁡9(x)3+3cos⁡7(x)7−cos⁡5(x)5\frac{\cos^{11}{\left(x \right)}}{11} - \frac{\cos^{9}{\left(x \right)}}{3} + \frac{3 \cos^{7}{\left(x \right)}}{7} - \frac{\cos^{5}{\left(x \right)}}{5}

        Method #3

        1. Rewrite the integrand:

          (1−cos⁡2(x))3sin⁡(x)cos⁡4(x)=−sin⁡(x)cos⁡10(x)+3sin⁡(x)cos⁡8(x)−3sin⁡(x)cos⁡6(x)+sin⁡(x)cos⁡4(x)\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{4}{\left(x \right)} = - \sin{\left(x \right)} \cos^{10}{\left(x \right)} + 3 \sin{\left(x \right)} \cos^{8}{\left(x \right)} - 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)} + \sin{\left(x \right)} \cos^{4}{\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⁡(x)cos⁡10(x)) dx=−∫sin⁡(x)cos⁡10(x) dx\int \left(- \sin{\left(x \right)} \cos^{10}{\left(x \right)}\right)\, dx = - \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 u^{10}\, du

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

                ∫(−u10) du=−∫u10 du\int \left(- u^{10}\right)\, 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: cos⁡11(x)11\frac{\cos^{11}{\left(x \right)}}{11}

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

            ∫3sin⁡(x)cos⁡8(x) dx=3∫sin⁡(x)cos⁡8(x) dx\int 3 \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx = 3 \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 u^{8}\, du

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

                ∫(−u8) du=−∫u8 du\int \left(- u^{8}\right)\, 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: −cos⁡9(x)3- \frac{\cos^{9}{\left(x \right)}}{3}

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

            ∫(−3sin⁡(x)cos⁡6(x)) dx=−3∫sin⁡(x)cos⁡6(x) dx\int \left(- 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - 3 \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 u^{6}\, du

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

                ∫(−u6) du=−∫u6 du\int \left(- u^{6}\right)\, 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: 3cos⁡7(x)7\frac{3 \cos^{7}{\left(x \right)}}{7}

          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 u^{4}\, du

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

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

          The result is: cos⁡11(x)11−cos⁡9(x)3+3cos⁡7(x)7−cos⁡5(x)5\frac{\cos^{11}{\left(x \right)}}{11} - \frac{\cos^{9}{\left(x \right)}}{3} + \frac{3 \cos^{7}{\left(x \right)}}{7} - \frac{\cos^{5}{\left(x \right)}}{5}

      So, the result is: −512cos⁡11(x)11+512cos⁡9(x)3−1536cos⁡7(x)7+512cos⁡5(x)5- \frac{512 \cos^{11}{\left(x \right)}}{11} + \frac{512 \cos^{9}{\left(x \right)}}{3} - \frac{1536 \cos^{7}{\left(x \right)}}{7} + \frac{512 \cos^{5}{\left(x \right)}}{5}

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

      ∫384sin⁡7(x)cos⁡2(x) dx=384∫sin⁡7(x)cos⁡2(x) dx\int 384 \sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 384 \int \sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx

      1. Rewrite the integrand:

        sin⁡7(x)cos⁡2(x)=(1−cos⁡2(x))3sin⁡(x)cos⁡2(x)\sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)}

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

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

        ∫(u8−3u6+3u4−u2) du\int \left(u^{8} - 3 u^{6} + 3 u^{4} - u^{2}\right)\, du

        1. Integrate term-by-term:

          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}

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

            ∫(−3u6) du=−3∫u6 du\int \left(- 3 u^{6}\right)\, du = - 3 \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: −3u77- \frac{3 u^{7}}{7}

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

            ∫3u4 du=3∫u4 du\int 3 u^{4}\, du = 3 \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: 3u55\frac{3 u^{5}}{5}

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

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

          The result is: u99−3u77+3u55−u33\frac{u^{9}}{9} - \frac{3 u^{7}}{7} + \frac{3 u^{5}}{5} - \frac{u^{3}}{3}

        Now substitute uu back in:

        cos⁡9(x)9−3cos⁡7(x)7+3cos⁡5(x)5−cos⁡3(x)3\frac{\cos^{9}{\left(x \right)}}{9} - \frac{3 \cos^{7}{\left(x \right)}}{7} + \frac{3 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}

      So, the result is: 128cos⁡9(x)3−1152cos⁡7(x)7+1152cos⁡5(x)5−128cos⁡3(x)\frac{128 \cos^{9}{\left(x \right)}}{3} - \frac{1152 \cos^{7}{\left(x \right)}}{7} + \frac{1152 \cos^{5}{\left(x \right)}}{5} - 128 \cos^{3}{\left(x \right)}

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

      ∫768sin⁡5(x)cos⁡4(x) dx=768∫sin⁡5(x)cos⁡4(x) dx\int 768 \sin^{5}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 768 \int \sin^{5}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx

      1. Rewrite the integrand:

        sin⁡5(x)cos⁡4(x)=(1−cos⁡2(x))2sin⁡(x)cos⁡4(x)\sin^{5}{\left(x \right)} \cos^{4}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos^{4}{\left(x \right)}

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

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

        ∫(−u8+2u6−u4) du\int \left(- u^{8} + 2 u^{6} - u^{4}\right)\, du

        1. Integrate term-by-term:

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

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

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

            ∫2u6 du=2∫u6 du\int 2 u^{6}\, du = 2 \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: 2u77\frac{2 u^{7}}{7}

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

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

          The result is: −u99+2u77−u55- \frac{u^{9}}{9} + \frac{2 u^{7}}{7} - \frac{u^{5}}{5}

        Now substitute uu back in:

        −cos⁡9(x)9+2cos⁡7(x)7−cos⁡5(x)5- \frac{\cos^{9}{\left(x \right)}}{9} + \frac{2 \cos^{7}{\left(x \right)}}{7} - \frac{\cos^{5}{\left(x \right)}}{5}

      So, the result is: −256cos⁡9(x)3+1536cos⁡7(x)7−768cos⁡5(x)5- \frac{256 \cos^{9}{\left(x \right)}}{3} + \frac{1536 \cos^{7}{\left(x \right)}}{7} - \frac{768 \cos^{5}{\left(x \right)}}{5}

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

      ∫(−576sin⁡5(x)cos⁡2(x)) dx=−576∫sin⁡5(x)cos⁡2(x) dx\int \left(- 576 \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 576 \int \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx

      1. Rewrite the integrand:

        sin⁡5(x)cos⁡2(x)=(1−cos⁡2(x))2sin⁡(x)cos⁡2(x)\sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)}

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

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

        ∫(−u6+2u4−u2) du\int \left(- u^{6} + 2 u^{4} - u^{2}\right)\, du

        1. Integrate term-by-term:

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

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

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

            ∫2u4 du=2∫u4 du\int 2 u^{4}\, du = 2 \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: 2u55\frac{2 u^{5}}{5}

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

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

          The result is: −u77+2u55−u33- \frac{u^{7}}{7} + \frac{2 u^{5}}{5} - \frac{u^{3}}{3}

        Now substitute uu back in:

        −cos⁡7(x)7+2cos⁡5(x)5−cos⁡3(x)3- \frac{\cos^{7}{\left(x \right)}}{7} + \frac{2 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}

      So, the result is: 576cos⁡7(x)7−1152cos⁡5(x)5+192cos⁡3(x)\frac{576 \cos^{7}{\left(x \right)}}{7} - \frac{1152 \cos^{5}{\left(x \right)}}{5} + 192 \cos^{3}{\left(x \right)}

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

      ∫(−320sin⁡3(x)cos⁡4(x)) dx=−320∫sin⁡3(x)cos⁡4(x) dx\int \left(- 320 \sin^{3}{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 320 \int \sin^{3}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx

      1. Rewrite the integrand:

        sin⁡3(x)cos⁡4(x)=(1−cos⁡2(x))sin⁡(x)cos⁡4(x)\sin^{3}{\left(x \right)} \cos^{4}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{4}{\left(x \right)}

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

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

        ∫(u6−u4) du\int \left(u^{6} - u^{4}\right)\, du

        1. Integrate term-by-term:

          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}

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

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

          The result is: u77−u55\frac{u^{7}}{7} - \frac{u^{5}}{5}

        Now substitute uu back in:

        cos⁡7(x)7−cos⁡5(x)5\frac{\cos^{7}{\left(x \right)}}{7} - \frac{\cos^{5}{\left(x \right)}}{5}

      So, the result is: −320cos⁡7(x)7+64cos⁡5(x)- \frac{320 \cos^{7}{\left(x \right)}}{7} + 64 \cos^{5}{\left(x \right)}

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

      ∫240sin⁡3(x)cos⁡2(x) dx=240∫sin⁡3(x)cos⁡2(x) dx\int 240 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 240 \int \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx

      1. Rewrite the integrand:

        sin⁡3(x)cos⁡2(x)=(1−cos⁡2(x))sin⁡(x)cos⁡2(x)\sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)}

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

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

        ∫(u4−u2) du\int \left(u^{4} - u^{2}\right)\, du

        1. Integrate term-by-term:

          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}

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

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

          The result is: u55−u33\frac{u^{5}}{5} - \frac{u^{3}}{3}

        Now substitute uu back in:

        cos⁡5(x)5−cos⁡3(x)3\frac{\cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}

      So, the result is: 48cos⁡5(x)−80cos⁡3(x)48 \cos^{5}{\left(x \right)} - 80 \cos^{3}{\left(x \right)}

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

      ∫32sin⁡(x)cos⁡4(x) dx=32∫sin⁡(x)cos⁡4(x) dx\int 32 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 32 \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 u^{4}\, du

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

          ∫(−u4) du=−∫u4 du\int \left(- u^{4}\right)\, 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: −32cos⁡5(x)5- \frac{32 \cos^{5}{\left(x \right)}}{5}

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

      ∫(−24sin⁡(x)cos⁡2(x)) dx=−24∫sin⁡(x)cos⁡2(x) dx\int \left(- 24 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 24 \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 u^{2}\, du

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

          ∫(−u2) du=−∫u2 du\int \left(- u^{2}\right)\, 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: 8cos⁡3(x)8 \cos^{3}{\left(x \right)}

    The result is: −512cos⁡11(x)11+128cos⁡9(x)−128cos⁡7(x)+272cos⁡5(x)5−8cos⁡3(x)- \frac{512 \cos^{11}{\left(x \right)}}{11} + 128 \cos^{9}{\left(x \right)} - 128 \cos^{7}{\left(x \right)} + \frac{272 \cos^{5}{\left(x \right)}}{5} - 8 \cos^{3}{\left(x \right)}

  3. Now simplify:

    8(−320cos⁡8(x)+880cos⁡6(x)−880cos⁡4(x)+374cos⁡2(x)−55)cos⁡3(x)55\frac{8 \left(- 320 \cos^{8}{\left(x \right)} + 880 \cos^{6}{\left(x \right)} - 880 \cos^{4}{\left(x \right)} + 374 \cos^{2}{\left(x \right)} - 55\right) \cos^{3}{\left(x \right)}}{55}

  4. Add the constant of integration:

    8(−320cos⁡8(x)+880cos⁡6(x)−880cos⁡4(x)+374cos⁡2(x)−55)cos⁡3(x)55+constant\frac{8 \left(- 320 \cos^{8}{\left(x \right)} + 880 \cos^{6}{\left(x \right)} - 880 \cos^{4}{\left(x \right)} + 374 \cos^{2}{\left(x \right)} - 55\right) \cos^{3}{\left(x \right)}}{55}+ \mathrm{constant}

The answer is:

8(−320cos⁡8(x)+880cos⁡6(x)−880cos⁡4(x)+374cos⁡2(x)−55)cos⁡3(x)55+constant\frac{8 \left(- 320 \cos^{8}{\left(x \right)} + 880 \cos^{6}{\left(x \right)} - 880 \cos^{4}{\left(x \right)} + 374 \cos^{2}{\left(x \right)} - 55\right) \cos^{3}{\left(x \right)}}{55}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                          11             5   
 |                                   7           3             9      512*cos  (x)   272*cos (x)
 | sin(8*x)*cos(3*x) dx = C - 128*cos (x) - 8*cos (x) + 128*cos (x) - ------------ + -----------
 |                                                                         11             5     
/
−cos⁡(11 x)22−cos⁡(5 x)10-{{\cos \left(11\,x\right)}\over{22}}-{{\cos \left(5\,x\right) }\over{10}}
Simplify
The answer [src] 
8    8*cos(3)*cos(8)   3*sin(3)*sin(8)
-- - --------------- - ---------------
55          55                55
855−5 cos⁡11+11 cos⁡5110{{8}\over{55}}-{{5\,\cos 11+11\,\cos 5}\over{110}} 
=
= 
8    8*cos(3)*cos(8)   3*sin(3)*sin(8)
-- - --------------- - ---------------
55          55                55
−8cos⁡(3)cos⁡(8)55−3sin⁡(3)sin⁡(8)55+855- \frac{8 \cos{\left(3 \right)} \cos{\left(8 \right)}}{55} - \frac{3 \sin{\left(3 \right)} \sin{\left(8 \right)}}{55} + \frac{8}{55}
Simplify
Numerical answer [src] 
0.116887158817857
0.116887158817857
The graph
Integral of sin8x*cos3x dx

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

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