Integral of sin²x*cosx dx

The solution

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 pi                  
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6

$$\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{2}} \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$$

Integral(sin(x)^2*cos(x), (x, pi/6, pi/2))

Detail solution

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

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

$\int u^{2}\, du$
1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int u^{2}\, du = \frac{u^{3}}{3}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = C + \frac{\sin^{3}{\left(x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

7/24

$$\frac{7}{24}$$

7/24

$$\frac{7}{24}$$

7/24

Numerical answer [src]

0.291666666666667

0.291666666666667

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

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Integral of sin²x*cosx dx

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The solution