Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x^e
Integral of √(x+4)
Integral of sqrt(1+x^4)
Integral of sin(2x)cos(2x)
Identical expressions
cos(x/ two)-x+pi
co sinus of e of (x divide by 2) minus x plus Pi
co sinus of e of (x divide by two) minus x plus Pi
cosx/2-x+pi
cos(x divide by 2)-x+pi
cos(x/2)-x+pidx
Similar expressions
cos(x/2)-x-pi
cos(x/2)+x+pi

Integral of cos(x/2)-x+pi dx

The solution

You have entered [src]

 pi                     
  /                     
 |                      
 |  /   /x\         \   
 |  |cos|-| - x + pi| dx
 |  \   \2/         /   
 |                      
/                       
0

$$\int\limits_{0}^{\pi} \left(\left(- x + \cos{\left(\frac{x}{2} \right)}\right) + \pi\right)\, dx$$

Integral(cos(x/2) - x + pi, (x, 0, pi))

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- x\right)\, dx = - \int x\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $- \frac{x^{2}}{2}$
  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:
    
    $2 \sin{\left(\frac{x}{2} \right)}$
  The result is: $- \frac{x^{2}}{2} + 2 \sin{\left(\frac{x}{2} \right)}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \pi\, dx = \pi x$
The result is: $- \frac{x^{2}}{2} + \pi x + 2 \sin{\left(\frac{x}{2} \right)}$
Now simplify:

$- \frac{x^{2}}{2} + \pi x + 2 \sin{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$- \frac{x^{2}}{2} + \pi x + 2 \sin{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$- \frac{x^{2}}{2} + \pi x + 2 \sin{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$2 + \frac{\pi^{2}}{2}$$

$$2 + \frac{\pi^{2}}{2}$$

2 + pi^2/2

Numerical answer [src]

6.93480220054468

6.93480220054468

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of cos(x/2)-x+pi dx

Limits of integration:

The graph:

Piecewise:

The solution