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Resolución de integrales/ (x+y)/ x^(2)/ (x+y)/(1+x^(2)+y^(2))

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |     x + y      
 |  ----------- dx
 |       2    2   
 |  1 + x  + y    
 |                
/                 
0
∫01x+yy2+(x2+1) dx\int\limits_{0}^{1} \frac{x + y}{y^{2} + \left(x^{2} + 1\right)}\, dx 
Integral((x + y)/(1 + x^2 + y^2), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    x+yy2+(x2+1)=xx2+y2+1+yx2+y2+1\frac{x + y}{y^{2} + \left(x^{2} + 1\right)} = \frac{x}{x^{2} + y^{2} + 1} + \frac{y}{x^{2} + y^{2} + 1}

  2. Integramos término a término:

    1. La integral del producto de una función por una constante es la constante por la integral de esta función:

      ∫xx2+y2+1 dx=∫2xx2+y2+1 dx2\int \frac{x}{x^{2} + y^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + y^{2} + 1}\, dx}{2}

      1. que u=x2+y2+1u = x^{2} + y^{2} + 1.

        Luego que du=2xdxdu = 2 x dx y ponemos du2\frac{du}{2}:

        ∫12u du\int \frac{1}{2 u}\, du

        1. Integral 1u\frac{1}{u} es log⁡(u)\log{\left(u \right)}.

        Si ahora sustituir uu más en:

        log⁡(x2+y2+1)\log{\left(x^{2} + y^{2} + 1 \right)}

      Por lo tanto, el resultado es: log⁡(x2+y2+1)2\frac{\log{\left(x^{2} + y^{2} + 1 \right)}}{2}

    1. La integral del producto de una función por una constante es la constante por la integral de esta función:

      ∫yx2+y2+1 dx=y∫1x2+y2+1 dx\int \frac{y}{x^{2} + y^{2} + 1}\, dx = y \int \frac{1}{x^{2} + y^{2} + 1}\, dx

      1. Integral 1x2+1\frac{1}{x^{2} + 1} es atan⁡(xy2+1)y2+1\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2} + 1}} \right)}}{\sqrt{y^{2} + 1}}.

      Por lo tanto, el resultado es: yatan⁡(xy2+1)y2+1\frac{y \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2} + 1}} \right)}}{\sqrt{y^{2} + 1}}

    El resultado es: yatan⁡(xy2+1)y2+1+log⁡(x2+y2+1)2\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}

  3. Añadimos la constante de integración:

    yatan⁡(xy2+1)y2+1+log⁡(x2+y2+1)2+constant\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}

Respuesta:

yatan⁡(xy2+1)y2+1+log⁡(x2+y2+1)2+constant\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}

Respuesta (Indefinida) [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      
/
∫x+yy2+(x2+1) dx=C+yatan⁡(xy2+1)y2+1+log⁡(x2+y2+1)2\int \frac{x + y}{y^{2} + \left(x^{2} + 1\right)}\, dx = C + \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}
Simplificar
Respuesta [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−y2−12(y2+1)+12)log⁡(2y2(−y−y2−12(y2+1)+12)−y2−y−y2−1y2+1y)+(−y−y2−12(y2+1)+12)log⁡(1+2y2(−y−y2−12(y2+1)+12)−y2−y−y2−1y2+1y)−(y−y2−12(y2+1)+12)log⁡(2y2(y−y2−12(y2+1)+12)−y2+y−y2−1y2+1y)+(y−y2−12(y2+1)+12)log⁡(1+2y2(y−y2−12(y2+1)+12)−y2+y−y2−1y2+1y)- \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)} 
=
= 
                        /                /         _________\        _________\                           /                /         _________\        _________\                           /            /         _________\        _________\                           /            /         _________\        _________\
                        |                |        /       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−y2−12(y2+1)+12)log⁡(2y2(−y−y2−12(y2+1)+12)−y2−y−y2−1y2+1y)+(−y−y2−12(y2+1)+12)log⁡(1+2y2(−y−y2−12(y2+1)+12)−y2−y−y2−1y2+1y)−(y−y2−12(y2+1)+12)log⁡(2y2(y−y2−12(y2+1)+12)−y2+y−y2−1y2+1y)+(y−y2−12(y2+1)+12)log⁡(1+2y2(y−y2−12(y2+1)+12)−y2+y−y2−1y2+1y)- \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)} 
(1/2 + y*sqrt(-1 - y^2)/(2*(1 + y^2)))*log(1 + (-y^2 + 2*y^2*(1/2 + y*sqrt(-1 - y^2)/(2*(1 + y^2))) + y*sqrt(-1 - y^2)/(1 + y^2))/y) + (1/2 - y*sqrt(-1 - y^2)/(2*(1 + y^2)))*log(1 + (-y^2 + 2*y^2*(1/2 - y*sqrt(-1 - y^2)/(2*(1 + y^2))) - y*sqrt(-1 - y^2)/(1 + y^2))/y) - (1/2 + y*sqrt(-1 - y^2)/(2*(1 + y^2)))*log((-y^2 + 2*y^2*(1/2 + y*sqrt(-1 - y^2)/(2*(1 + y^2))) + y*sqrt(-1 - y^2)/(1 + y^2))/y) - (1/2 - y*sqrt(-1 - y^2)/(2*(1 + y^2)))*log((-y^2 + 2*y^2*(1/2 - y*sqrt(-1 - y^2)/(2*(1 + y^2))) - y*sqrt(-1 - y^2)/(1 + y^2))/y)

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

    © Sr Examen – Калькулятор