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Solving integrals/ sinx/(1+3cosx)

Integral of sinx/(1+3cosx) dx

The solution

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

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

Simplify

Detail solution

Let $u = 3 \cos{\left(x \right)} + 1$ .

Then let $du = - 3 \sin{\left(x \right)} dx$ and substitute $- \frac{du}{3}$ :

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

$- \frac{\log{\left(3 \cos{\left(x \right)} + 1 \right)}}{3}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                       
 |                                        
 |    sin(x)             log(1 + 3*cos(x))
 | ------------ dx = C - -----------------
 | 1 + 3*cos(x)                  3        
 |                                        
/

$$-{{\log \left(3\,\cos x+1\right)}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Simplify

Numerical answer [src]

0.0

0.0

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 sinx/(1+3cosx) dx

Limits of integration:

The graph:

Piecewise:

The solution