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Integral of cos(3*x-pi/2) dx

The solution

You have entered [src]

  x                 
  /                 
 |                  
 |     /      pi\   
 |  cos|3*x - --| dx
 |     \      2 /   
 |                  
/                   
0

$$\int\limits_{0}^{x} \cos{\left(3 x - \frac{\pi}{2} \right)}\, dx$$

Integral(cos(3*x - pi/2), (x, 0, x))

Detail solution

Let $u = 3 x - \frac{\pi}{2}$ .

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:

$\frac{\sin{\left(3 x - \frac{\pi}{2} \right)}}{3}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                          /      pi\
 |                        sin|3*x - --|
 |    /      pi\             \      2 /
 | cos|3*x - --| dx = C + -------------
 |    \      2 /                3      
 |                                     
/

$$\int \cos{\left(3 x - \frac{\pi}{2} \right)}\, dx = C + \frac{\sin{\left(3 x - \frac{\pi}{2} \right)}}{3}$$

Simplify

The answer [src]

1   cos(3*x)
- - --------
3      3

$$\frac{1}{3} - \frac{\cos{\left(3 x \right)}}{3}$$

1   cos(3*x)
- - --------
3      3

$$\frac{1}{3} - \frac{\cos{\left(3 x \right)}}{3}$$

1/3 - cos(3*x)/3

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

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Integral of cos(3*x-pi/2) dx

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Piecewise:

The solution