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Integral of d{x}:
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Identical expressions
(x+y)/(one +x^(two)+y^(two))
(x plus y) divide by (1 plus x to the power of (2) plus y to the power of (2))
(x plus y) divide by (one plus x to the power of (two) plus y to the power of (two))
(x+y)/(1+x(2)+y(2))
x+y/1+x2+y2
x+y/1+x^2+y^2
(x+y) divide by (1+x^(2)+y^(2))
(x+y)/(1+x^(2)+y^(2))dx
Similar expressions
(x+y)/(1+x^(2)-y^(2))
(x+y)/(1-x^(2)+y^(2))
(x-y)/(1+x^(2)+y^(2))

Solving integrals/ (x+y)/(1+x^(2)+y^(2))

Integral of (x+y)/(1+x^(2)+y^(2)) dx

The solution

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

$$\int\limits_{0}^{1} \frac{x + y}{x^{2} + y^{2} + 1}\, dx$$

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

Detail solution

Rewrite the integrand:

$\frac{x + y}{x^{2} + y^{2} + 1} = \frac{x}{x^{2} + y^{2} + 1} + \frac{y}{x^{2} + y^{2} + 1}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x}{x^{2} + y^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + y^{2} + 1}\, dx}{2}$
  1. Let $u = x^{2} + y^{2} + 1$ .
    
    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:
    
    $\log{\left(x^{2} + y^{2} + 1 \right)}$
  So, the result is: $\frac{\log{\left(x^{2} + y^{2} + 1 \right)}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{y}{x^{2} + y^{2} + 1}\, dx = y \int \frac{1}{x^{2} + y^{2} + 1}\, dx$
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2} + 1}} \right)}}{\sqrt{y^{2} + 1}}$ .
  So, the result is: $\frac{y \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2} + 1}} \right)}}{\sqrt{y^{2} + 1}}$
The result is: $\frac{y \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2} + 1}} \right)}}{\sqrt{y^{2} + 1}} + \frac{\log{\left(x^{2} + y^{2} + 1 \right)}}{2}$
Add the constant of integration:

$\frac{y \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2} + 1}} \right)}}{\sqrt{y^{2} + 1}} + \frac{\log{\left(x^{2} + y^{2} + 1 \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{y \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2} + 1}} \right)}}{\sqrt{y^{2} + 1}} + \frac{\log{\left(x^{2} + y^{2} + 1 \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                                 /     x     \
                                           y*atan|-----------|
  /                                              |   ________|
 |                         /     2    2\         |  /      2 |
 |    x + y             log\1 + x  + y /         \\/  1 + y  /
 | ----------- dx = C + ---------------- + -------------------
 |      2    2                 2                  ________    
 | 1 + x  + y                                    /      2     
 |                                             \/  1 + y      
/

$${{y\,\arctan \left({{x}\over{\sqrt{y^2+1}}}\right)}\over{\sqrt{y^2+ 1}}}+{{\log \left(y^2+x^2+1\right)}\over{2}}$$

Simplify

The answer [src]

                        /                /         _________\        _________\                           /                /         _________\        _________\                           /            /         _________\        _________\                           /            /         _________\        _________\
                        |                |        /       2 |       /       2 |                           |                |        /       2 |       /       2 |                           |            |        /       2 |       /       2 |                           |            |        /       2 |       /       2 |
                        |       2      2 |1   y*\/  -1 - y  |   y*\/  -1 - y  |                           |       2      2 |1   y*\/  -1 - y  |   y*\/  -1 - y  |                           |   2      2 |1   y*\/  -1 - y  |   y*\/  -1 - y  |                           |   2      2 |1   y*\/  -1 - y  |   y*\/  -1 - y  |
/         _________\    |    - y  + 2*y *|- + --------------| + --------------|   /         _________\    |    - y  + 2*y *|- - --------------| - --------------|   /         _________\    |- y  + 2*y *|- + --------------| + --------------|   /         _________\    |- y  + 2*y *|- - --------------| - --------------|
|        /       2 |    |                |2       /     2\  |            2    |   |        /       2 |    |                |2       /     2\  |            2    |   |        /       2 |    |            |2       /     2\  |            2    |   |        /       2 |    |            |2       /     2\  |            2    |
|1   y*\/  -1 - y  |    |                \      2*\1 + y /  /       1 + y     |   |1   y*\/  -1 - y  |    |                \      2*\1 + y /  /       1 + y     |   |1   y*\/  -1 - y  |    |            \      2*\1 + y /  /       1 + y     |   |1   y*\/  -1 - y  |    |            \      2*\1 + y /  /       1 + y     |
|- + --------------|*log|1 + -------------------------------------------------| + |- - --------------|*log|1 + -------------------------------------------------| - |- + --------------|*log|-------------------------------------------------| - |- - --------------|*log|-------------------------------------------------|
|2       /     2\  |    \                            y                        /   |2       /     2\  |    \                            y                        /   |2       /     2\  |    \                        y                        /   |2       /     2\  |    \                        y                        /
\      2*\1 + y /  /                                                              \      2*\1 + y /  /                                                              \      2*\1 + y /  /                                                          \      2*\1 + y /  /

$${{y\,\arctan \left({{1}\over{\sqrt{y^2+1}}}\right)}\over{\sqrt{y^2+ 1}}}+{{y^2\,\log \left(y^2+2\right)}\over{2\,y^2+2}}+{{\log \left(y^ 2+2\right)}\over{2\,y^2+2}}-{{y^2\,\log \left(y^2+1\right)}\over{2\, y^2+2}}-{{\log \left(y^2+1\right)}\over{2\,y^2+2}}$$

                        /                /         _________\        _________\                           /                /         _________\        _________\                           /            /         _________\        _________\                           /            /         _________\        _________\
                        |                |        /       2 |       /       2 |                           |                |        /       2 |       /       2 |                           |            |        /       2 |       /       2 |                           |            |        /       2 |       /       2 |
                        |       2      2 |1   y*\/  -1 - y  |   y*\/  -1 - y  |                           |       2      2 |1   y*\/  -1 - y  |   y*\/  -1 - y  |                           |   2      2 |1   y*\/  -1 - y  |   y*\/  -1 - y  |                           |   2      2 |1   y*\/  -1 - y  |   y*\/  -1 - y  |
/         _________\    |    - y  + 2*y *|- + --------------| + --------------|   /         _________\    |    - y  + 2*y *|- - --------------| - --------------|   /         _________\    |- y  + 2*y *|- + --------------| + --------------|   /         _________\    |- y  + 2*y *|- - --------------| - --------------|
|        /       2 |    |                |2       /     2\  |            2    |   |        /       2 |    |                |2       /     2\  |            2    |   |        /       2 |    |            |2       /     2\  |            2    |   |        /       2 |    |            |2       /     2\  |            2    |
|1   y*\/  -1 - y  |    |                \      2*\1 + y /  /       1 + y     |   |1   y*\/  -1 - y  |    |                \      2*\1 + y /  /       1 + y     |   |1   y*\/  -1 - y  |    |            \      2*\1 + y /  /       1 + y     |   |1   y*\/  -1 - y  |    |            \      2*\1 + y /  /       1 + y     |
|- + --------------|*log|1 + -------------------------------------------------| + |- - --------------|*log|1 + -------------------------------------------------| - |- + --------------|*log|-------------------------------------------------| - |- - --------------|*log|-------------------------------------------------|
|2       /     2\  |    \                            y                        /   |2       /     2\  |    \                            y                        /   |2       /     2\  |    \                        y                        /   |2       /     2\  |    \                        y                        /
\      2*\1 + y /  /                                                              \      2*\1 + y /  /                                                              \      2*\1 + y /  /                                                          \      2*\1 + y /  /

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

Simplify

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

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Integral of (x+y)/(1+x^(2)+y^(2)) dx

Limits of integration:

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Piecewise:

The solution