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Solving integrals/ 1/2cos(x+(pi/4))

Integral of 1/2cos(x+(pi/4)) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 pi               
 --               
 4                
  /               
 |                
 |     /    pi\   
 |  cos|x + --|   
 |     \    4 /   
 |  ----------- dx
 |       2        
 |                
/                 
0
$$\int\limits_{0}^{\frac{\pi}{4}} \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{2}\, dx$$ 
Integral(cos(x + pi/4)/2, (x, 0, pi/4))
Detail solution

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

    1. Let .

      Then let and substitute :

      1. The integral of cosine is sine:

      Now substitute back in:

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                                
 |                                 
 |    /    pi\             /    pi\
 | cos|x + --|          sin|x + --|
 |    \    4 /             \    4 /
 | ----------- dx = C + -----------
 |      2                    2     
 |                                 
/
$$\int \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{2}\, dx = C + \frac{\sin{\left(x + \frac{\pi}{4} \right)}}{2}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
      ___
1   \/ 2 
- - -----
2     4
$$\frac{1}{2} - \frac{\sqrt{2}}{4}$$ 
=
= 
      ___
1   \/ 2 
- - -----
2     4
$$\frac{1}{2} - \frac{\sqrt{2}}{4}$$ 
1/2 - sqrt(2)/4
Numerical answer [src] 
0.146446609406726
0.146446609406726

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

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