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