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Solving integrals/ sin10x*sin7x

Integral of sin10x*sin7x dx

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∫00sin⁡(7x)sin⁡(10x) dx\int\limits_{0}^{0} \sin{\left(7 x \right)} \sin{\left(10 x \right)}\, dx 
Integral(sin(10*x)*sin(7*x), (x, 0, 0))
Detail solution

  1. Rewrite the integrand:

    sin⁡(7x)sin⁡(10x)=−32768sin⁡16(x)cos⁡(x)+122880sin⁡14(x)cos⁡(x)−186368sin⁡12(x)cos⁡(x)+146432sin⁡10(x)cos⁡(x)−63360sin⁡8(x)cos⁡(x)+14784sin⁡6(x)cos⁡(x)−1680sin⁡4(x)cos⁡(x)+70sin⁡2(x)cos⁡(x)\sin{\left(7 x \right)} \sin{\left(10 x \right)} = - 32768 \sin^{16}{\left(x \right)} \cos{\left(x \right)} + 122880 \sin^{14}{\left(x \right)} \cos{\left(x \right)} - 186368 \sin^{12}{\left(x \right)} \cos{\left(x \right)} + 146432 \sin^{10}{\left(x \right)} \cos{\left(x \right)} - 63360 \sin^{8}{\left(x \right)} \cos{\left(x \right)} + 14784 \sin^{6}{\left(x \right)} \cos{\left(x \right)} - 1680 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 70 \sin^{2}{\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:

      ∫(−32768sin⁡16(x)cos⁡(x)) dx=−32768∫sin⁡16(x)cos⁡(x) dx\int \left(- 32768 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 32768 \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: −32768sin⁡17(x)17- \frac{32768 \sin^{17}{\left(x \right)}}{17}

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

      ∫122880sin⁡14(x)cos⁡(x) dx=122880∫sin⁡14(x)cos⁡(x) dx\int 122880 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx = 122880 \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: 8192sin⁡15(x)8192 \sin^{15}{\left(x \right)}

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

      ∫(−186368sin⁡12(x)cos⁡(x)) dx=−186368∫sin⁡12(x)cos⁡(x) dx\int \left(- 186368 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 186368 \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: −14336sin⁡13(x)- 14336 \sin^{13}{\left(x \right)}

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

      ∫146432sin⁡10(x)cos⁡(x) dx=146432∫sin⁡10(x)cos⁡(x) dx\int 146432 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx = 146432 \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: 13312sin⁡11(x)13312 \sin^{11}{\left(x \right)}

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

      ∫(−63360sin⁡8(x)cos⁡(x)) dx=−63360∫sin⁡8(x)cos⁡(x) dx\int \left(- 63360 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 63360 \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: −7040sin⁡9(x)- 7040 \sin^{9}{\left(x \right)}

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

      ∫14784sin⁡6(x)cos⁡(x) dx=14784∫sin⁡6(x)cos⁡(x) dx\int 14784 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx = 14784 \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: 2112sin⁡7(x)2112 \sin^{7}{\left(x \right)}

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

      ∫(−1680sin⁡4(x)cos⁡(x)) dx=−1680∫sin⁡4(x)cos⁡(x) dx\int \left(- 1680 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1680 \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: −336sin⁡5(x)- 336 \sin^{5}{\left(x \right)}

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

      ∫70sin⁡2(x)cos⁡(x) dx=70∫sin⁡2(x)cos⁡(x) dx\int 70 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = 70 \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: 70sin⁡3(x)3\frac{70 \sin^{3}{\left(x \right)}}{3}

    The result is: −32768sin⁡17(x)17+8192sin⁡15(x)−14336sin⁡13(x)+13312sin⁡11(x)−7040sin⁡9(x)+2112sin⁡7(x)−336sin⁡5(x)+70sin⁡3(x)3- \frac{32768 \sin^{17}{\left(x \right)}}{17} + 8192 \sin^{15}{\left(x \right)} - 14336 \sin^{13}{\left(x \right)} + 13312 \sin^{11}{\left(x \right)} - 7040 \sin^{9}{\left(x \right)} + 2112 \sin^{7}{\left(x \right)} - 336 \sin^{5}{\left(x \right)} + \frac{70 \sin^{3}{\left(x \right)}}{3}

  3. Now simplify:

    2(−49152sin⁡14(x)+208896sin⁡12(x)−365568sin⁡10(x)+339456sin⁡8(x)−179520sin⁡6(x)+53856sin⁡4(x)−8568sin⁡2(x)+595)sin⁡3(x)51\frac{2 \left(- 49152 \sin^{14}{\left(x \right)} + 208896 \sin^{12}{\left(x \right)} - 365568 \sin^{10}{\left(x \right)} + 339456 \sin^{8}{\left(x \right)} - 179520 \sin^{6}{\left(x \right)} + 53856 \sin^{4}{\left(x \right)} - 8568 \sin^{2}{\left(x \right)} + 595\right) \sin^{3}{\left(x \right)}}{51}

  4. Add the constant of integration:

    2(−49152sin⁡14(x)+208896sin⁡12(x)−365568sin⁡10(x)+339456sin⁡8(x)−179520sin⁡6(x)+53856sin⁡4(x)−8568sin⁡2(x)+595)sin⁡3(x)51+constant\frac{2 \left(- 49152 \sin^{14}{\left(x \right)} + 208896 \sin^{12}{\left(x \right)} - 365568 \sin^{10}{\left(x \right)} + 339456 \sin^{8}{\left(x \right)} - 179520 \sin^{6}{\left(x \right)} + 53856 \sin^{4}{\left(x \right)} - 8568 \sin^{2}{\left(x \right)} + 595\right) \sin^{3}{\left(x \right)}}{51}+ \mathrm{constant}

The answer is:

2(−49152sin⁡14(x)+208896sin⁡12(x)−365568sin⁡10(x)+339456sin⁡8(x)−179520sin⁡6(x)+53856sin⁡4(x)−8568sin⁡2(x)+595)sin⁡3(x)51+constant\frac{2 \left(- 49152 \sin^{14}{\left(x \right)} + 208896 \sin^{12}{\left(x \right)} - 365568 \sin^{10}{\left(x \right)} + 339456 \sin^{8}{\left(x \right)} - 179520 \sin^{6}{\left(x \right)} + 53856 \sin^{4}{\left(x \right)} - 8568 \sin^{2}{\left(x \right)} + 595\right) \sin^{3}{\left(x \right)}}{51}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                                                                                   17            3   
 |                                      13              9             5              7              15               11      32768*sin  (x)   70*sin (x)
 | sin(10*x)*sin(7*x) dx = C - 14336*sin  (x) - 7040*sin (x) - 336*sin (x) + 2112*sin (x) + 8192*sin  (x) + 13312*sin  (x) - -------------- + ----------
 |                                                                                                                                 17             3     
/
∫sin⁡(7x)sin⁡(10x) dx=C−32768sin⁡17(x)17+8192sin⁡15(x)−14336sin⁡13(x)+13312sin⁡11(x)−7040sin⁡9(x)+2112sin⁡7(x)−336sin⁡5(x)+70sin⁡3(x)3\int \sin{\left(7 x \right)} \sin{\left(10 x \right)}\, dx = C - \frac{32768 \sin^{17}{\left(x \right)}}{17} + 8192 \sin^{15}{\left(x \right)} - 14336 \sin^{13}{\left(x \right)} + 13312 \sin^{11}{\left(x \right)} - 7040 \sin^{9}{\left(x \right)} + 2112 \sin^{7}{\left(x \right)} - 336 \sin^{5}{\left(x \right)} + \frac{70 \sin^{3}{\left(x \right)}}{3}
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