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Integral of sinxcos^2*x dx

The solution

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  x                  
  -                  
  2                  
  /                  
 |                   
 |            2      
 |  sin(x)*cos (x) dx
 |                   
/                    
1

$$\int\limits_{1}^{\frac{x}{2}} \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

Integral(sin(x)*cos(x)^2, (x, 1, x/2))

Detail solution

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

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

$\int \left(- u^{2}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int u^{2}\, du = - \int u^{2}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int u^{2}\, du = \frac{u^{3}}{3}$
  So, the result is: $- \frac{u^{3}}{3}$
Now substitute $u$ back in:

$- \frac{\cos^{3}{\left(x \right)}}{3}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                               
 |                            3   
 |           2             cos (x)
 | sin(x)*cos (x) dx = C - -------
 |                            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

$$- \frac{\cos^{3}{\left(\frac{x}{2} \right)}}{3} + \frac{\cos^{3}{\left(1 \right)}}{3}$$

     3/x\          
  cos |-|      3   
      \2/   cos (1)
- ------- + -------
     3         3

$$- \frac{\cos^{3}{\left(\frac{x}{2} \right)}}{3} + \frac{\cos^{3}{\left(1 \right)}}{3}$$

-cos(x/2)^3/3 + cos(1)^3/3

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

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Integral of sinxcos^2*x dx

Limits of integration:

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The solution