Integral of cos(3x+2) dx

The solution

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  1                
  /                
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 |  cos(3*x + 2) dx
 |                 
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0

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

Integral(cos(3*x + 2), (x, 0, 1))

Detail solution

Let $u = 3 x + 2$ .

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:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                  
 |                       sin(3*x + 2)
 | cos(3*x + 2) dx = C + ------------
 |                            3      
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  sin(2)   sin(5)
- ------ + ------
    3        3

$$\frac{\sin{\left(5 \right)}}{3} - \frac{\sin{\left(2 \right)}}{3}$$

  sin(2)   sin(5)
- ------ + ------
    3        3

$$\frac{\sin{\left(5 \right)}}{3} - \frac{\sin{\left(2 \right)}}{3}$$

-sin(2)/3 + sin(5)/3

Numerical answer [src]

-0.62274056716294

-0.62274056716294

The graph

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

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Identical expressions

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Integral of cos(3x+2) dx

Limits of integration:

The graph:

Piecewise:

The solution