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Solving integrals/ (cos⁵x)½

Integral of (cos⁵x)½ dx

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  1           
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 |     5      
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 |  ------- dx
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∫01cos⁡5(x)2 dx\int\limits_{0}^{1} \frac{\cos^{5}{\left(x \right)}}{2}\, dx 
Integral(cos(x)^5/2, (x, 0, 1))
Detail solution

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

    ∫cos⁡5(x)2 dx=∫cos⁡5(x) dx2\int \frac{\cos^{5}{\left(x \right)}}{2}\, dx = \frac{\int \cos^{5}{\left(x \right)}\, dx}{2}

    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)}

    So, the result is: sin⁡5(x)10−sin⁡3(x)3+sin⁡(x)2\frac{\sin^{5}{\left(x \right)}}{10} - \frac{\sin^{3}{\left(x \right)}}{3} + \frac{\sin{\left(x \right)}}{2}

  2. Add the constant of integration:

    sin⁡5(x)10−sin⁡3(x)3+sin⁡(x)2+constant\frac{\sin^{5}{\left(x \right)}}{10} - \frac{\sin^{3}{\left(x \right)}}{3} + \frac{\sin{\left(x \right)}}{2}+ \mathrm{constant}

The answer is:

sin⁡5(x)10−sin⁡3(x)3+sin⁡(x)2+constant\frac{\sin^{5}{\left(x \right)}}{10} - \frac{\sin^{3}{\left(x \right)}}{3} + \frac{\sin{\left(x \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                           
 |                                            
 |    5                         3         5   
 | cos (x)          sin(x)   sin (x)   sin (x)
 | ------- dx = C + ------ - ------- + -------
 |    2               2         3         10  
 |                                            
/
∫cos⁡5(x)2 dx=C+sin⁡5(x)10−sin⁡3(x)3+sin⁡(x)2\int \frac{\cos^{5}{\left(x \right)}}{2}\, dx = C + \frac{\sin^{5}{\left(x \right)}}{10} - \frac{\sin^{3}{\left(x \right)}}{3} + \frac{\sin{\left(x \right)}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
            3         5   
sin(1)   sin (1)   sin (1)
------ - ------- + -------
  2         3         10
−sin⁡3(1)3+sin⁡5(1)10+sin⁡(1)2- \frac{\sin^{3}{\left(1 \right)}}{3} + \frac{\sin^{5}{\left(1 \right)}}{10} + \frac{\sin{\left(1 \right)}}{2} 
=
= 
            3         5   
sin(1)   sin (1)   sin (1)
------ - ------- + -------
  2         3         10
−sin⁡3(1)3+sin⁡5(1)10+sin⁡(1)2- \frac{\sin^{3}{\left(1 \right)}}{3} + \frac{\sin^{5}{\left(1 \right)}}{10} + \frac{\sin{\left(1 \right)}}{2} 
sin(1)/2 - sin(1)^3/3 + sin(1)^5/10
Numerical answer [src] 
0.264316406455608
0.264316406455608

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

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