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(1/2)cos(x-pi/4)dx

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

The solution

You have entered [src]

  4               
  /               
 |                
 |     /    pi\   
 |  cos|x + --|   
 |     \    4 /   
 |  ----------- dx
 |       2        
 |                
/                 
1

$$\int\limits_{1}^{4} \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{2}\, dx$$

Integral(cos(x + pi/4)/2, (x, 1, 4))

Detail solution

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

$\int \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{2}\, dx = \frac{\int \cos{\left(x + \frac{\pi}{4} \right)}\, dx}{2}$
1. Let $u = x + \frac{\pi}{4}$ .
  
  Then let $du = dx$ and substitute $du$ :
  
  $\int \cos{\left(u \right)}\, du$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $\sin{\left(x + \frac{\pi}{4} \right)}$
So, the result is: $\frac{\sin{\left(x + \frac{\pi}{4} \right)}}{2}$
Now simplify:

$\frac{\sin{\left(x + \frac{\pi}{4} \right)}}{2}$
Add the constant of integration:

$\frac{\sin{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\sin{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}$

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\      /    pi\
sin|4 + --|   sin|1 + --|
   \    4 /      \    4 /
----------- - -----------
     2             2

$$\frac{\sin{\left(\frac{\pi}{4} + 4 \right)}}{2} - \frac{\sin{\left(\frac{\pi}{4} + 1 \right)}}{2}$$

   /    pi\      /    pi\
sin|4 + --|   sin|1 + --|
   \    4 /      \    4 /
----------- - -----------
     2             2

$$\frac{\sin{\left(\frac{\pi}{4} + 4 \right)}}{2} - \frac{\sin{\left(\frac{\pi}{4} + 1 \right)}}{2}$$

sin(4 + pi/4)/2 - sin(1 + pi/4)/2

Numerical answer [src]

-0.9871986385713

-0.9871986385713

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

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

Limits of integration:

The graph:

Piecewise:

The solution