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Solving integrals/ 0.5*cos(x+pi/4)

Integral of 0.5*cos(x+pi/4) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  4               
  /               
 |                
 |     /    pi\   
 |  cos|x + --|   
 |     \    4 /   
 |  ----------- dx
 |       2        
 |                
/                 
0
$$\int\limits_{0}^{4} \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{2}\, dx$$ 
Integral(cos(x + pi/4)/2, (x, 0, 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] 
   /    pi\        
sin|4 + --|     ___
   \    4 /   \/ 2 
----------- - -----
     2          4
$$\frac{\sin{\left(\frac{\pi}{4} + 4 \right)}}{2} - \frac{\sqrt{2}}{4}$$ 
=
= 
   /    pi\        
sin|4 + --|     ___
   \    4 /   \/ 2 
----------- - -----
     2          4
$$\frac{\sin{\left(\frac{\pi}{4} + 4 \right)}}{2} - \frac{\sqrt{2}}{4}$$ 
sin(4 + pi/4)/2 - sqrt(2)/4
Numerical answer [src] 
-0.852221397214836
-0.852221397214836

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

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