Mister Exam

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Solving integrals/ 2*cos^2(z)*sin(z)

Integral of 2cos^2(z)sin(z) dz

The solution

You have entered [src]

 pi                    
 --                    
 2                     
  /                    
 |                     
 |       2             
 |  2*cos (z)*sin(z) dz
 |                     
/                      
0

$$\int\limits_{0}^{\frac{\pi}{2}} 2 \sin{\left(z \right)} \cos^{2}{\left(z \right)}\, dz$$

Simplify

Detail solution

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

$\int 2 \sin{\left(z \right)} \cos^{2}{\left(z \right)}\, dz = 2 \int \sin{\left(z \right)} \cos^{2}{\left(z \right)}\, dz$
1. Let $u = \cos{\left(z \right)}$ .
  
  Then let $du = - \sin{\left(z \right)} dz$ 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:
  
  $- \frac{\cos^{3}{\left(z \right)}}{3}$
So, the result is: $- \frac{2 \cos^{3}{\left(z \right)}}{3}$
Add the constant of integration:

$- \frac{2 \cos^{3}{\left(z \right)}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{2 \cos^{3}{\left(z \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                3   
 |      2                    2*cos (z)
 | 2*cos (z)*sin(z) dz = C - ---------
 |                               3    
/

$$-{{2\,\cos ^3z}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2/3

$$\frac{2}{3}$$

2/3

$$\frac{2}{3}$$

Simplify

Numerical answer [src]

0.666666666666667

0.666666666666667

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of 2cos^2(z)sin(z) dz

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Integral of 2*cos^2(z)*sin(z) dz

Limits of integration:

The graph:

Piecewise:

The solution

Integral of 2cos^2(z)sin(z) dz