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Solving integrals/ sin^4(x)cos(x)

Integral of sin^4(x)cos(x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 pi                  
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 |     4             
 |  sin (x)*cos(x) dx
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pi                   
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6
∫π6π3sin⁡4(x)cos⁡(x) dx\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx 
Integral(sin(x)^4*cos(x), (x, pi/6, pi/3))
Detail solution

  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}

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                               
 |                            5   
 |    4                    sin (x)
 | sin (x)*cos(x) dx = C + -------
 |                            5   
/
∫sin⁡4(x)cos⁡(x) dx=C+sin⁡5(x)5\int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = C + \frac{\sin^{5}{\left(x \right)}}{5}
Simplify
The graph
0.550.600.650.700.750.800.850.900.951.000.00.5
Plot the graph f(x)
The answer [src] 
            ___
   1    9*\/ 3 
- --- + -------
  160     160
−1160+93160- \frac{1}{160} + \frac{9 \sqrt{3}}{160} 
=
= 
            ___
   1    9*\/ 3 
- --- + -------
  160     160
−1160+93160- \frac{1}{160} + \frac{9 \sqrt{3}}{160} 
-1/160 + 9*sqrt(3)/160
Numerical answer [src] 
0.0911778579257494
0.0911778579257494

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

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