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Solving integrals/ sinx*cos^2xdx

Integral of sinx*cos^2xdx dx

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The graph:

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The solution

You have entered [src] 
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∫012sin⁡(x)cos⁡2(x) dx\int\limits_{0}^{\frac{1}{2}} \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx 
Integral(sin(x)*cos(x)^2, (x, 0, 1/2))
Detail solution

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

    Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

    ∫(−u2) du\int \left(- u^{2}\right)\, du

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

      ∫u2 du=−∫u2 du\int u^{2}\, 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}

      So, the result is: −u33- \frac{u^{3}}{3}

    Now substitute uu back in:

    −cos⁡3(x)3- \frac{\cos^{3}{\left(x \right)}}{3}

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                               
 |                            3   
 |           2             cos (x)
 | sin(x)*cos (x) dx = C - -------
 |                            3   
/
∫sin⁡(x)cos⁡2(x) dx=C−cos⁡3(x)3\int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = C - \frac{\cos^{3}{\left(x \right)}}{3}
Simplify
The graph
0.000.500.050.100.150.200.250.300.350.400.451.0-1.0
Plot the graph f(x)
The answer [src] 
       3     
1   cos (1/2)
- - ---------
3       3
13−cos⁡3(12)3\frac{1}{3} - \frac{\cos^{3}{\left(\frac{1}{2} \right)}}{3} 
=
= 
       3     
1   cos (1/2)
- - ---------
3       3
13−cos⁡3(12)3\frac{1}{3} - \frac{\cos^{3}{\left(\frac{1}{2} \right)}}{3} 
1/3 - cos(1/2)^3/3
Numerical answer [src] 
0.108042926055098
0.108042926055098

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

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