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cos²xsinx
Solving integrals/ cos²xsinx

Integral of cos²xsinx dx

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The solution

You have entered [src] 
  1                  
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 |     2             
 |  cos (x)*sin(x) dx
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0
∫01sin⁡(x)cos⁡2(x) dx\int\limits_{0}^{1} \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx
Simplify
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 u^{2}\, du

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

      ∫(−u2) du=−∫u2 du\int \left(- u^{2}\right)\, 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)
 | cos (x)*sin(x) dx = C - -------
 |                            3   
/
−cos⁡3x3-{{\cos ^3x}\over{3}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901.0-1.0
Plot the graph f(x)
The answer [src] 
       3   
1   cos (1)
- - -------
3      3
13−cos⁡313{{1}\over{3}}-{{\cos ^31}\over{3}} 
=
= 
       3   
1   cos (1)
- - -------
3      3
−cos⁡3(1)3+13- \frac{\cos^{3}{\left(1 \right)}}{3} + \frac{1}{3}
Simplify
Numerical answer [src] 
0.280757131583002
0.280757131583002
The graph
Integral of cos²xsinx dx

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

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