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cos(2x+(pi)/(four))
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cos(2x+(pi) divide by (4))
cos(2x+(pi)/(4))dx
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dx/cos(2*x+pi/4)^2
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Solving integrals/ cos(2x+(pi)/(4))

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

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = 2 x + \frac{\pi}{4}$ .

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

$\int \frac{\cos{\left(u \right)}}{4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
  So, the result is: $\frac{\sin{\left(u \right)}}{2}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                          /      pi\
 |                        sin|2*x + --|
 |    /      pi\             \      4 /
 | cos|2*x + --| dx = C + -------------
 |    \      4 /                2      
 |                                     
/

$${{\sin \left(2\,x+{{\pi}\over{4}}\right)}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   ___ 
-\/ 2  
-------
   2

$${{\sin \left({{5\,\pi}\over{4}}\right)}\over{2}}-{{\sin \left({{9\, \pi}\over{4}}\right)}\over{2}}$$

   ___ 
-\/ 2  
-------
   2

$$- \frac{\sqrt{2}}{2}$$

Simplify

Numerical answer [src]

-0.707106781186547

-0.707106781186547

The graph

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution