Sr Examen
Integral de 1/(1+x^2+y^2)^2 dx
Solución
Ha introducido
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1 / | | 1 | -------------- dx | 2 | / 2 2\ | \1 + x + y / | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(y^{2} + \left(x^{2} + 1\right)\right)^{2}}\, dx$$
Integral(1/((1 + x^2 + y^2)^2), (x, 0, 1))
Respuesta (Indefinida)
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___________ / ___________ ___________ ___________\ ___________ / ___________ ___________ ___________\
/ -1 | / -1 4 / -1 2 / -1 | / -1 | / -1 4 / -1 2 / -1 |
/ --------- *log|x - / --------- - y * / --------- - 2*y * / --------- | / --------- *log|x + / --------- + y * / --------- + 2*y * / --------- |
/ / 3 | / 3 / 3 / 3 | / 3 | / 3 / 3 / 3 |
| / / 2\ | / / 2\ / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ / / 2\ |
| 1 x \/ \1 + y / \ \/ \1 + y / \/ \1 + y / \/ \1 + y / / \/ \1 + y / \ \/ \1 + y / \/ \1 + y / \/ \1 + y / /
| -------------- dx = C + ------------------------------- - -------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------
| 2 4 2 2 / 2\ 4 4
| / 2 2\ 2 + 2*y + 4*y + x *\2 + 2*y /
| \1 + x + y /
|
/
$$\int \frac{1}{\left(y^{2} + \left(x^{2} + 1\right)\right)^{2}}\, dx = C + \frac{x}{x^{2} \left(2 y^{2} + 2\right) + 2 y^{4} + 4 y^{2} + 2} - \frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(x - y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \right)}}{4} + \frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(x + y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \right)}}{4}$$
Respuesta
[src]
___________ / ___________ ___________ ___________\ ___________ / ___________ ___________ ___________\ ___________ / ___________ ___________ ___________\ ___________ / ___________ ___________ ___________\
/ -1 | / -1 4 / -1 2 / -1 | / -1 | / -1 4 / -1 2 / -1 | / -1 | / -1 4 / -1 2 / -1 | / -1 | / -1 4 / -1 2 / -1 |
/ --------- *log| / --------- + y * / --------- + 2*y * / --------- | / --------- *log|1 - / --------- - y * / --------- - 2*y * / --------- | / --------- *log|- / --------- - y * / --------- - 2*y * / --------- | / --------- *log|1 + / --------- + y * / --------- + 2*y * / --------- |
/ 3 | / 3 / 3 / 3 | / 3 | / 3 / 3 / 3 | / 3 | / 3 / 3 / 3 | / 3 | / 3 / 3 / 3 |
/ / 2\ | / / 2\ / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ / / 2\ |
1 \/ \1 + y / \\/ \1 + y / \/ \1 + y / \/ \1 + y / / \/ \1 + y / \ \/ \1 + y / \/ \1 + y / \/ \1 + y / / \/ \1 + y / \ \/ \1 + y / \/ \1 + y / \/ \1 + y / / \/ \1 + y / \ \/ \1 + y / \/ \1 + y / \/ \1 + y / /
--------------- - ---------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------
4 2 4 4 4 4
4 + 2*y + 6*y
$$\frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(- y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \right)}}{4} - \frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \right)}}{4} - \frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(- y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + 1 \right)}}{4} + \frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + 1 \right)}}{4} + \frac{1}{2 y^{4} + 6 y^{2} + 4}$$
=
=
___________ / ___________ ___________ ___________\ ___________ / ___________ ___________ ___________\ ___________ / ___________ ___________ ___________\ ___________ / ___________ ___________ ___________\
/ -1 | / -1 4 / -1 2 / -1 | / -1 | / -1 4 / -1 2 / -1 | / -1 | / -1 4 / -1 2 / -1 | / -1 | / -1 4 / -1 2 / -1 |
/ --------- *log| / --------- + y * / --------- + 2*y * / --------- | / --------- *log|1 - / --------- - y * / --------- - 2*y * / --------- | / --------- *log|- / --------- - y * / --------- - 2*y * / --------- | / --------- *log|1 + / --------- + y * / --------- + 2*y * / --------- |
/ 3 | / 3 / 3 / 3 | / 3 | / 3 / 3 / 3 | / 3 | / 3 / 3 / 3 | / 3 | / 3 / 3 / 3 |
/ / 2\ | / / 2\ / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ / / 2\ |
1 \/ \1 + y / \\/ \1 + y / \/ \1 + y / \/ \1 + y / / \/ \1 + y / \ \/ \1 + y / \/ \1 + y / \/ \1 + y / / \/ \1 + y / \ \/ \1 + y / \/ \1 + y / \/ \1 + y / / \/ \1 + y / \ \/ \1 + y / \/ \1 + y / \/ \1 + y / /
--------------- - ---------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------
4 2 4 4 4 4
4 + 2*y + 6*y
$$\frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(- y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \right)}}{4} - \frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \right)}}{4} - \frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(- y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} - \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + 1 \right)}}{4} + \frac{\sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} \log{\left(y^{4} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + 2 y^{2} \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + \sqrt{- \frac{1}{\left(y^{2} + 1\right)^{3}}} + 1 \right)}}{4} + \frac{1}{2 y^{4} + 6 y^{2} + 4}$$
1/(4 + 2*y^4 + 6*y^2) - sqrt(-1/(1 + y^2)^3)*log(sqrt(-1/(1 + y^2)^3) + y^4*sqrt(-1/(1 + y^2)^3) + 2*y^2*sqrt(-1/(1 + y^2)^3))/4 - sqrt(-1/(1 + y^2)^3)*log(1 - sqrt(-1/(1 + y^2)^3) - y^4*sqrt(-1/(1 + y^2)^3) - 2*y^2*sqrt(-1/(1 + y^2)^3))/4 + sqrt(-1/(1 + y^2)^3)*log(-sqrt(-1/(1 + y^2)^3) - y^4*sqrt(-1/(1 + y^2)^3) - 2*y^2*sqrt(-1/(1 + y^2)^3))/4 + sqrt(-1/(1 + y^2)^3)*log(1 + sqrt(-1/(1 + y^2)^3) + y^4*sqrt(-1/(1 + y^2)^3) + 2*y^2*sqrt(-1/(1 + y^2)^3))/4
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.