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cos^7(x)
Solving integrals/ cos^7(x)

Integral of cos^7(x) dx

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∫01cos⁡7(x) dx\int\limits_{0}^{1} \cos^{7}{\left(x \right)}\, dx 
Integral(cos(x)^7, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

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

  2. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

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

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

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

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

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

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

    Method #2

    1. Rewrite the integrand:

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

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

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

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

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

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

  3. Now simplify:

    (−sin⁡6(x)7+3sin⁡4(x)5+cos⁡2(x))sin⁡(x)\left(- \frac{\sin^{6}{\left(x \right)}}{7} + \frac{3 \sin^{4}{\left(x \right)}}{5} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}

  4. Add the constant of integration:

    (−sin⁡6(x)7+3sin⁡4(x)5+cos⁡2(x))sin⁡(x)+constant\left(- \frac{\sin^{6}{\left(x \right)}}{7} + \frac{3 \sin^{4}{\left(x \right)}}{5} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}+ \mathrm{constant}

The answer is:

(−sin⁡6(x)7+3sin⁡4(x)5+cos⁡2(x))sin⁡(x)+constant\left(- \frac{\sin^{6}{\left(x \right)}}{7} + \frac{3 \sin^{4}{\left(x \right)}}{5} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                       
 |                               7           5            
 |    7                3      sin (x)   3*sin (x)         
 | cos (x) dx = C - sin (x) - ------- + --------- + sin(x)
 |                               7          5             
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−sin⁡7x7+3 sin⁡5x5−sin⁡3x+sin⁡x-{{\sin ^7x}\over{7}}+{{3\,\sin ^5x}\over{5}}-\sin ^3x+\sin x
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
               7           5            
     3      sin (1)   3*sin (1)         
- sin (1) - ------- + --------- + sin(1)
               7          5
−5 sin⁡71−21 sin⁡51+35 sin⁡31−35 sin⁡135-{{5\,\sin ^71-21\,\sin ^51+35\,\sin ^31-35\,\sin 1}\over{35}} 
=
= 
               7           5            
     3      sin (1)   3*sin (1)         
- sin (1) - ------- + --------- + sin(1)
               7          5
−sin⁡3(1)−sin⁡7(1)7+3sin⁡5(1)5+sin⁡(1)- \sin^{3}{\left(1 \right)} - \frac{\sin^{7}{\left(1 \right)}}{7} + \frac{3 \sin^{5}{\left(1 \right)}}{5} + \sin{\left(1 \right)}
Simplify
Numerical answer [src] 
0.456104465133679
0.456104465133679
The graph
Integral of cos^7(x) dx

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

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