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Integral of cos(3/2x-1)dx dx

The solution

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  1                
  /                
 |                 
 |     /3*x    \   
 |  cos|--- - 1| dx
 |     \ 2     /   
 |                 
/                  
0

$$\int\limits_{0}^{1} \cos{\left(\frac{3 x}{2} - 1 \right)}\, dx$$

Integral(cos(3*x/2 - 1), (x, 0, 1))

Detail solution

Let $u = \frac{3 x}{2} - 1$ .

Then let $du = \frac{3 dx}{2}$ and substitute $\frac{2 du}{3}$ :

$\int \frac{2 \cos{\left(u \right)}}{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \cos{\left(u \right)}\, du = \frac{2 \int \cos{\left(u \right)}\, du}{3}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
  So, the result is: $\frac{2 \sin{\left(u \right)}}{3}$
Now substitute $u$ back in:

$\frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}$
Now simplify:

$\frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}$
Add the constant of integration:

$\frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                           /3*x    \
 |                       2*sin|--- - 1|
 |    /3*x    \               \ 2     /
 | cos|--- - 1| dx = C + --------------
 |    \ 2     /                3       
 |                                     
/

$$\int \cos{\left(\frac{3 x}{2} - 1 \right)}\, dx = C + \frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2*sin(1)   2*sin(1/2)
-------- + ----------
   3           3

$$\frac{2 \sin{\left(\frac{1}{2} \right)}}{3} + \frac{2 \sin{\left(1 \right)}}{3}$$

2*sin(1)   2*sin(1/2)
-------- + ----------
   3           3

$$\frac{2 \sin{\left(\frac{1}{2} \right)}}{3} + \frac{2 \sin{\left(1 \right)}}{3}$$

2*sin(1)/3 + 2*sin(1/2)/3

Numerical answer [src]

0.880597682274733

0.880597682274733

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

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Identical expressions

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Integral of cos(3/2x-1)dx dx

Limits of integration:

The graph:

Piecewise:

The solution