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cos^5(x)
Solving integrals/ cos^5(x)

Integral of cos^5(x) dx

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∫01cos⁡5(x) dx\int\limits_{0}^{1} \cos^{5}{\left(x \right)}\, dx 
Integral(cos(x)^5, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

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

  2. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

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

    2. Integrate term-by-term:

      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}

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

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

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

    Method #2

    1. Rewrite the integrand:

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

    2. Integrate term-by-term:

      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}

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

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

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                             
 |                       3         5            
 |    5             2*sin (x)   sin (x)         
 | cos (x) dx = C - --------- + ------- + sin(x)
 |                      3          5            
/
∫cos⁡5(x) dx=C+sin⁡5(x)5−2sin⁡3(x)3+sin⁡(x)\int \cos^{5}{\left(x \right)}\, dx = C + \frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
       3         5            
  2*sin (1)   sin (1)         
- --------- + ------- + sin(1)
      3          5
−2sin⁡3(1)3+sin⁡5(1)5+sin⁡(1)- \frac{2 \sin^{3}{\left(1 \right)}}{3} + \frac{\sin^{5}{\left(1 \right)}}{5} + \sin{\left(1 \right)} 
=
= 
       3         5            
  2*sin (1)   sin (1)         
- --------- + ------- + sin(1)
      3          5
−2sin⁡3(1)3+sin⁡5(1)5+sin⁡(1)- \frac{2 \sin^{3}{\left(1 \right)}}{3} + \frac{\sin^{5}{\left(1 \right)}}{5} + \sin{\left(1 \right)} 
-2*sin(1)^3/3 + sin(1)^5/5 + sin(1)
Numerical answer [src] 
0.528632812911216
0.528632812911216
The graph
Integral of cos^5(x) dx

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

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