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