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Similar expressions
sinxcosx/sin^2x-1

Integral of sinxcosx/sin^2x+1 dx

The solution

You have entered [src]

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

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

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

Detail solution

Integrate term-by-term:
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(\sin{\left(x \right)} \right)}$
  Method #2
  1. Let $u = \sin^{2}{\left(x \right)}$ .
    
    Then let $du = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ 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:
      
      $\int \frac{1}{2 u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
      
      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:
    
    $\frac{\log{\left(\sin^{2}{\left(x \right)} \right)}}{2}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $x + \log{\left(\sin{\left(x \right)} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\log \sin x+x$$

Simplify

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Simplify

Numerical answer [src]

44.9178423877238

44.9178423877238

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

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Identical expressions

Similar expressions

Integral of sinxcosx/sin^2x+1 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2