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(1/2)*x*cos(x/2)dx

Integral of (1/2)xcos(x/2) dx

The solution

You have entered [src]

 pi            
  /            
 |             
 |  x    /x\   
 |  -*cos|-| dx
 |  2    \2/   
 |             
/              
-oo

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

Integral((x/2)*cos(x/2), (x, -oo, pi))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \frac{x}{2}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ .

Then $\operatorname{du}{\left(x \right)} = \frac{1}{2}$ .

To find $v{\left(x \right)}$ :
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
  
  $\int 2 \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $2 \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $2 \sin{\left(\frac{x}{2} \right)}$
Now evaluate the sub-integral.
Let $u = \frac{x}{2}$ .

Then let $du = \frac{dx}{2}$ and substitute $2 du$ :

$\int 2 \sin{\left(u \right)}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = 2 \int \sin{\left(u \right)}\, du$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- 2 \cos{\left(u \right)}$
Now substitute $u$ back in:

$- 2 \cos{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$x \sin{\left(\frac{x}{2} \right)} + 2 \cos{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$x \sin{\left(\frac{x}{2} \right)} + 2 \cos{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 | x    /x\               /x\        /x\
 | -*cos|-| dx = C + 2*cos|-| + x*sin|-|
 | 2    \2/               \2/        \2/
 |                                      
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

<-oo, oo>

$$\left\langle -\infty, \infty\right\rangle$$

<-oo, oo>

$$\left\langle -\infty, \infty\right\rangle$$

AccumBounds(-oo, oo)

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

Other calculators

How to use it?

Identical expressions

Integral of (1/2)xcos(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Integral of (1/2)*x*cos(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of (1/2)xcos(x/2) dx