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Solving integrals/ sinxcos^2x

Integral of sinxcos^2x dx

The solution

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 67                  
 --                  
 10                  
  /                  
 |                   
 |            2      
 |  sin(x)*cos (x) dx
 |                   
/                    
0

$$\int\limits_{0}^{\frac{67}{10}} \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

Integral(sin(x)*cos(x)^2, (x, 0, 67/10))

Detail solution

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

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

$\int u^{2}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
  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   
/

$$-{{\cos ^3x}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       3/67\
    cos |--|
1       \10/
- - --------
3      3

$${{1}\over{3}}-{{\cos ^3\left({{67}\over{10}}\right)}\over{3}}$$

       3/67\
    cos |--|
1       \10/
- - --------
3      3

$$- \frac{\cos^{3}{\left(\frac{67}{10} \right)}}{3} + \frac{1}{3}$$

Simplify

Numerical answer [src]

0.0784958039671756

0.0784958039671756

The graph

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

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Identical expressions

Integral of sinxcos^2x dx

Limits of integration:

The graph:

Piecewise:

The solution