Mister Exam

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0,5*sin(x-y^2)

Integral of 0,5*sin(x+y^2) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |     /     2\   
 |  sin\x + y /   
 |  ----------- dx
 |       2        
 |                
/                 
0

$$\int\limits_{0}^{1} \frac{\sin{\left(x + y^{2} \right)}}{2}\, dx$$

Integral(sin(x + y^2)/2, (x, 0, 1))

Detail solution

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

$\int \frac{\sin{\left(x + y^{2} \right)}}{2}\, dx = \frac{\int \sin{\left(x + y^{2} \right)}\, dx}{2}$
1. Let $u = x + y^{2}$ .
  
  Then let $du = dx$ and substitute $du$ :
  
  $\int \sin{\left(u \right)}\, du$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- \cos{\left(x + y^{2} \right)}$
So, the result is: $- \frac{\cos{\left(x + y^{2} \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

   / 2\      /     2\
cos\y /   cos\1 + y /
------- - -----------
   2           2

$$\frac{\cos{\left(y^{2} \right)}}{2} - \frac{\cos{\left(y^{2} + 1 \right)}}{2}$$

   / 2\      /     2\
cos\y /   cos\1 + y /
------- - -----------
   2           2

$$\frac{\cos{\left(y^{2} \right)}}{2} - \frac{\cos{\left(y^{2} + 1 \right)}}{2}$$

cos(y^2)/2 - cos(1 + y^2)/2

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

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Similar expressions

Integral of 0,5*sin(x+y^2) dx

Limits of integration:

The graph:

Piecewise:

The solution