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