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Integral of (x+cosx)/(x^2+2sinx) dx

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = x^{2} + 2 \sin{\left(x \right)}$ .

Then let $du = \left(2 x + 2 \cos{\left(x \right)}\right) dx$ and substitute $\frac{du}{2}$ :

$\int \frac{1}{4 u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{2 u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  So, the result is: $\frac{\log{\left(u \right)}}{2}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$${{\log \left(2\,\sin x+x^2\right)}\over{2}}$$

Simplify

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Simplify

Numerical answer [src]

22.1921064478628

22.1921064478628

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

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Identical expressions

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Integral of (x+cosx)/(x^2+2sinx) dx

Limits of integration:

The graph:

Piecewise:

The solution