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Solving integrals/ sin^7(x)

Integral of sin^7(x) dx

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The solution

You have entered [src] 
 POST_GRBEK_SMALL_pi          
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∫0POSTGRBEKSMALLπsin⁡7(x) dx\int\limits_{0}^{POST_{GRBEK SMALL \pi}} \sin^{7}{\left(x \right)}\, dx 
Integral(sin(x)^7, (x, 0, POST_GRBEK_SMALL_pi))
Detail solution

  1. Rewrite the integrand:

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

  2. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      (1−cos⁡2(x))3sin⁡(x)=−sin⁡(x)cos⁡6(x)+3sin⁡(x)cos⁡4(x)−3sin⁡(x)cos⁡2(x)+sin⁡(x)\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} = - \sin{\left(x \right)} \cos^{6}{\left(x \right)} + 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\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⁡6(x)) dx=−∫sin⁡(x)cos⁡6(x) dx\int \left(- \sin{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - \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: cos⁡7(x)7\frac{\cos^{7}{\left(x \right)}}{7}

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

        ∫3sin⁡(x)cos⁡4(x) dx=3∫sin⁡(x)cos⁡4(x) dx\int 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 3 \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: −3cos⁡5(x)5- \frac{3 \cos^{5}{\left(x \right)}}{5}

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

        ∫(−3sin⁡(x)cos⁡2(x)) dx=−3∫sin⁡(x)cos⁡2(x) dx\int \left(- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 3 \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: cos⁡3(x)\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⁡7(x)7−3cos⁡5(x)5+cos⁡3(x)−cos⁡(x)\frac{\cos^{7}{\left(x \right)}}{7} - \frac{3 \cos^{5}{\left(x \right)}}{5} + \cos^{3}{\left(x \right)} - \cos{\left(x \right)}

    Method #2

    1. Rewrite the integrand:

      (1−cos⁡2(x))3sin⁡(x)=−sin⁡(x)cos⁡6(x)+3sin⁡(x)cos⁡4(x)−3sin⁡(x)cos⁡2(x)+sin⁡(x)\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} = - \sin{\left(x \right)} \cos^{6}{\left(x \right)} + 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\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⁡6(x)) dx=−∫sin⁡(x)cos⁡6(x) dx\int \left(- \sin{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - \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: cos⁡7(x)7\frac{\cos^{7}{\left(x \right)}}{7}

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

        ∫3sin⁡(x)cos⁡4(x) dx=3∫sin⁡(x)cos⁡4(x) dx\int 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 3 \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: −3cos⁡5(x)5- \frac{3 \cos^{5}{\left(x \right)}}{5}

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

        ∫(−3sin⁡(x)cos⁡2(x)) dx=−3∫sin⁡(x)cos⁡2(x) dx\int \left(- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 3 \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: cos⁡3(x)\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⁡7(x)7−3cos⁡5(x)5+cos⁡3(x)−cos⁡(x)\frac{\cos^{7}{\left(x \right)}}{7} - \frac{3 \cos^{5}{\left(x \right)}}{5} + \cos^{3}{\left(x \right)} - \cos{\left(x \right)}

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                                       
 |                                          5         7   
 |    7                3               3*cos (x)   cos (x)
 | sin (x) dx = C + cos (x) - cos(x) - --------- + -------
 |                                         5          7   
/
cos⁡7x7−3 cos⁡5x5+cos⁡3x−cos⁡x{{\cos ^7x}\over{7}}-{{3\,\cos ^5x}\over{5}}+\cos ^3x-\cos x
Simplify
The answer [src] 
                                                                 5                           7                     
16      3                                                   3*cos (POST_GRBEK_SMALL_pi)   cos (POST_GRBEK_SMALL_pi)
-- + cos (POST_GRBEK_SMALL_pi) - cos(POST_GRBEK_SMALL_pi) - --------------------------- + -------------------------
35                                                                       5                            7
cos⁡7(POSTGRBEKSMALLπ)7−3cos⁡5(POSTGRBEKSMALLπ)5+cos⁡3(POSTGRBEKSMALLπ)−cos⁡(POSTGRBEKSMALLπ)+1635\frac{\cos^{7}{\left(POST_{GRBEK SMALL \pi} \right)}}{7} - \frac{3 \cos^{5}{\left(POST_{GRBEK SMALL \pi} \right)}}{5} + \cos^{3}{\left(POST_{GRBEK SMALL \pi} \right)} - \cos{\left(POST_{GRBEK SMALL \pi} \right)} + \frac{16}{35} 
=
= 
                                                                 5                           7                     
16      3                                                   3*cos (POST_GRBEK_SMALL_pi)   cos (POST_GRBEK_SMALL_pi)
-- + cos (POST_GRBEK_SMALL_pi) - cos(POST_GRBEK_SMALL_pi) - --------------------------- + -------------------------
35                                                                       5                            7
cos⁡7(POSTGRBEKSMALLπ)7−3cos⁡5(POSTGRBEKSMALLπ)5+cos⁡3(POSTGRBEKSMALLπ)−cos⁡(POSTGRBEKSMALLπ)+1635\frac{\cos^{7}{\left(POST_{GRBEK SMALL \pi} \right)}}{7} - \frac{3 \cos^{5}{\left(POST_{GRBEK SMALL \pi} \right)}}{5} + \cos^{3}{\left(POST_{GRBEK SMALL \pi} \right)} - \cos{\left(POST_{GRBEK SMALL \pi} \right)} + \frac{16}{35}
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