Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x*2^(-x)
Integral of ex^2
Integral of x^5/(x^2+1)
Integral of sin(x²)
Identical expressions
sin((x-y)/ two)
sinus of ((x minus y) divide by 2)
sinus of ((x minus y) divide by two)
sinx-y/2
sin((x-y) divide by 2)
sin((x-y)/2)dx
Similar expressions
sin((x+y)/2)

Integral of sin((x-y)/2) dy

The solution

You have entered [src]

 2*pi             
   /              
  |               
  |     /x - y\   
  |  sin|-----| dy
  |     \  2  /   
  |               
 /                
 0

$$\int\limits_{0}^{2 \pi} \sin{\left(\frac{x - y}{2} \right)}\, dy$$

Integral(sin((x - y)/2), (y, 0, 2*pi))

Detail solution

Let $u = \frac{x - y}{2}$ .

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

$\int \left(- 2 \sin{\left(u \right)}\right)\, du$
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:

$2 \cos{\left(\frac{x - y}{2} \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

      /x\
-4*cos|-|
      \2/

$$- 4 \cos{\left(\frac{x}{2} \right)}$$

      /x\
-4*cos|-|
      \2/

$$- 4 \cos{\left(\frac{x}{2} \right)}$$

-4*cos(x/2)

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sin((x-y)/2) dy

Limits of integration:

The graph:

Piecewise:

The solution