Integral of sin(3x+5) dx

The solution

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  1                
  /                
 |                 
 |  sin(3*x + 5) dx
 |                 
/                  
0

$$\int\limits_{0}^{1} \sin{\left(3 x + 5 \right)}\, dx$$

Simplify

Detail solution

Let $u = 3 x + 5$ .

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

$\int \frac{\sin{\left(u \right)}}{9}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sin{\left(u \right)}}{3}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{3}$
Now substitute $u$ back in:

$- \frac{\cos{\left(3 x + 5 \right)}}{3}$
Now simplify:

$- \frac{\cos{\left(3 x + 5 \right)}}{3}$
Add the constant of integration:

$- \frac{\cos{\left(3 x + 5 \right)}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(3 x + 5 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                       cos(3*x + 5)
 | sin(3*x + 5) dx = C - ------------
 |                            3      
/

$$-{{\cos \left(3\,x+5\right)}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  cos(8)   cos(5)
- ------ + ------
    3        3

$${{\cos 5-\cos 8}\over{3}}$$

  cos(8)   cos(5)
- ------ + ------
    3        3

$$- \frac{\cos{\left(8 \right)}}{3} + \frac{\cos{\left(5 \right)}}{3}$$

Simplify

Numerical answer [src]

0.143054073090613

0.143054073090613

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 sin(3x+5) dx

Limits of integration:

The graph:

Piecewise:

The solution