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Solving integrals/ sinxdx/(2+3cosx)

Integral of sinxdx/(2+3cosx) dx

The solution

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

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

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

Detail solution

Let $u = 3 \cos{\left(x \right)} + 2$ .

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:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                
 |                                                 
 |               1                log(2 + 3*cos(x))
 | sin(x)*1*------------ dx = C - -----------------
 |          2 + 3*cos(x)                  3        
 |                                                 
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  log(2 + 3*cos(1))   log(5)
- ----------------- + ------
          3             3

$$- \frac{\log{\left(3 \cos{\left(1 \right)} + 2 \right)}}{3} + \frac{\log{\left(5 \right)}}{3}$$

  log(2 + 3*cos(1))   log(5)
- ----------------- + ------
          3             3

$$- \frac{\log{\left(3 \cos{\left(1 \right)} + 2 \right)}}{3} + \frac{\log{\left(5 \right)}}{3}$$

Упростить

Numerical answer [src]

0.107571129416258

0.107571129416258

The graph

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

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Identical expressions

Similar expressions

Integral of sinxdx/(2+3cosx) dx

Limits of integration:

The graph:

Piecewise:

The solution