Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of (x+1)^2
Integral of 1/cos^2(3x)
Integral of xctg^2x
Integral of cos(3*x-pi/4)
Graphing y =:
cos(3*x-pi/4)
Derivative of:
cos(3*x-pi/4)
Identical expressions
cos(three *x-pi/ four)
co sinus of e of (3 multiply by x minus Pi divide by 4)
co sinus of e of (three multiply by x minus Pi divide by four)
cos(3x-pi/4)
cos3x-pi/4
cos(3*x-pi divide by 4)
cos(3*x-pi/4)dx
Similar expressions
cos(3*x+pi/4)

Integral of cos(3*x-pi/4) dx

The solution

You have entered [src]

 pi                 
  /                 
 |                  
 |     /      pi\   
 |  cos|3*x - --| dx
 |     \      4 /   
 |                  
/                   
pi                  
--                  
4

$$\int\limits_{\frac{\pi}{4}}^{\pi} \cos{\left(3 x - \frac{\pi}{4} \right)}\, dx$$

Integral(cos(3*x - pi/4), (x, pi/4, pi))

Detail solution

Let $u = 3 x - \frac{\pi}{4}$ .

Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :

$\int \frac{\cos{\left(u \right)}}{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
  So, the result is: $\frac{\sin{\left(u \right)}}{3}$
Now substitute $u$ back in:

$\frac{\sin{\left(3 x - \frac{\pi}{4} \right)}}{3}$
Now simplify:

$- \frac{\cos{\left(3 x + \frac{\pi}{4} \right)}}{3}$
Add the constant of integration:

$- \frac{\cos{\left(3 x + \frac{\pi}{4} \right)}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(3 x + \frac{\pi}{4} \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                          /      pi\
 |                        sin|3*x - --|
 |    /      pi\             \      4 /
 | cos|3*x - --| dx = C + -------------
 |    \      4 /                3      
 |                                     
/

$$\int \cos{\left(3 x - \frac{\pi}{4} \right)}\, dx = C + \frac{\sin{\left(3 x - \frac{\pi}{4} \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        ___
  1   \/ 2 
- - + -----
  3     6

$$- \frac{1}{3} + \frac{\sqrt{2}}{6}$$

        ___
  1   \/ 2 
- - + -----
  3     6

$$- \frac{1}{3} + \frac{\sqrt{2}}{6}$$

-1/3 + sqrt(2)/6

Numerical answer [src]

-0.0976310729378174

-0.0976310729378174

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of cos(3*x-pi/4) dx

Limits of integration:

The graph:

Piecewise:

The solution