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Solving integrals/ sinxcos^2*x

Integral of sinxcos^2*x dx

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The solution

You have entered [src] 
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∫1x2sin⁡(x)cos⁡2(x) dx\int\limits_{1}^{\frac{x}{2}} \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx 
Integral(sin(x)*cos(x)^2, (x, 1, x/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 answer [src] 
     3/x\          
  cos |-|      3   
      \2/   cos (1)
- ------- + -------
     3         3
−cos⁡3(x2)3+cos⁡3(1)3- \frac{\cos^{3}{\left(\frac{x}{2} \right)}}{3} + \frac{\cos^{3}{\left(1 \right)}}{3} 
=
= 
     3/x\          
  cos |-|      3   
      \2/   cos (1)
- ------- + -------
     3         3
−cos⁡3(x2)3+cos⁡3(1)3- \frac{\cos^{3}{\left(\frac{x}{2} \right)}}{3} + \frac{\cos^{3}{\left(1 \right)}}{3} 
-cos(x/2)^3/3 + cos(1)^3/3

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

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