Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x^(4/5)
Integral of (x^2)sinx
Integral of 6x+3
Integral of sin³xcosx
Identical expressions
x/(x^ two +y^ two)-y/x^ two
x divide by (x squared plus y squared ) minus y divide by x squared
x divide by (x to the power of two plus y to the power of two) minus y divide by x to the power of two
x/(x2+y2)-y/x2
x/x2+y2-y/x2
x/(x²+y²)-y/x²
x/(x to the power of 2+y to the power of 2)-y/x to the power of 2
x/x^2+y^2-y/x^2
x divide by (x^2+y^2)-y divide by x^2
x/(x^2+y^2)-y/x^2dx
Similar expressions
x/(x^2+y^2)+y/x^2
x/(x^2-y^2)-y/x^2

Integral of x/(x^2+y^2)-y/x^2 dx

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |  /   x      y \   
 |  |------- - --| dx
 |  | 2    2    2|   
 |  \x  + y    x /   
 |                   
/                    
0

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

Integral(x/(x^2 + y^2) - y/x^2, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x}{x^{2} + y^{2}}\, dx = \frac{\int \frac{2 x}{x^{2} + y^{2}}\, dx}{2}$
  1. Let $u = x^{2} + y^{2}$ .
    
    Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{2 u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(x^{2} + y^{2} \right)}$
  So, the result is: $\frac{\log{\left(x^{2} + y^{2} \right)}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{y}{x^{2}}\right)\, dx = - y \int \frac{1}{x^{2}}\, dx$
  So, the result is: $\text{NaN}$
The result is: $\text{NaN}$
Add the constant of integration:

$\text{NaN}+ \mathrm{constant}$

The answer is:

$\text{NaN}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                       
 |                        
 | /   x      y \         
 | |------- - --| dx = nan
 | | 2    2    2|         
 | \x  + y    x /         
 |                        
/

$$\int \left(\frac{x}{x^{2} + y^{2}} - \frac{y}{x^{2}}\right)\, dx = \text{NaN}$$

Simplify

The answer [src]

       /     2\             
    log\1 + y /             
y + ----------- - oo*sign(y)
         2

$$y + \frac{\log{\left(y^{2} + 1 \right)}}{2} - \infty \operatorname{sign}{\left(y \right)}$$

       /     2\             
    log\1 + y /             
y + ----------- - oo*sign(y)
         2

$$y + \frac{\log{\left(y^{2} + 1 \right)}}{2} - \infty \operatorname{sign}{\left(y \right)}$$

y + log(1 + y^2)/2 - oo*sign(y)

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of x/(x^2+y^2)-y/x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution