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sin^5xcosxdx
Solving integrals/ sin^5xcosxdx

Integral of sin^5xcosxdx dx

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The solution

You have entered [src] 
  1                    
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 |     5               
 |  sin (x)*cos(x)*1 dx
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0
∫01sin⁡5(x)cos⁡(x)1 dx\int\limits_{0}^{1} \sin^{5}{\left(x \right)} \cos{\left(x \right)} 1\, dx
Simplify
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

      Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

      ∫u5 du\int u^{5}\, du

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫u5 du=u66\int u^{5}\, du = \frac{u^{6}}{6}

      Now substitute uu back in:

      sin⁡6(x)6\frac{\sin^{6}{\left(x \right)}}{6}

    Method #2

    1. Rewrite the integrand:

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

    2. Let u=1−cos⁡2(x)u = 1 - \cos^{2}{\left(x \right)}.

      Then let du=2sin⁡(x)cos⁡(x)dxdu = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx and substitute du2\frac{du}{2}:

      ∫u24 du\int \frac{u^{2}}{4}\, du

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

        ∫u22 du=∫u2 du2\int \frac{u^{2}}{2}\, du = \frac{\int u^{2}\, du}{2}

        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: u36\frac{u^{3}}{6}

      Now substitute uu back in:

      (1−cos⁡2(x))36\frac{\left(1 - \cos^{2}{\left(x \right)}\right)^{3}}{6}

  2. Add the constant of integration:

    sin⁡6(x)6+constant\frac{\sin^{6}{\left(x \right)}}{6}+ \mathrm{constant}

The answer is:

sin⁡6(x)6+constant\frac{\sin^{6}{\left(x \right)}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 
 |                              6   
 |    5                      sin (x)
 | sin (x)*cos(x)*1 dx = C + -------
 |                              6   
/
sin⁡6x6{{\sin ^6x}\over{6}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.000.25
Plot the graph f(x)
The answer [src] 
   6   
sin (1)
-------
   6
sin⁡616{{\sin ^61}\over{6}} 
=
= 
   6   
sin (1)
-------
   6
sin⁡6(1)6\frac{\sin^{6}{\left(1 \right)}}{6}
Simplify
Numerical answer [src] 
0.0591675548769536
0.0591675548769536
The graph
Integral of sin^5xcosxdx dx

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

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