Sr Examen
Integral de (e^x*sqrt(e^x+2))/e^x+3 dx
Solución
Ha introducido
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pi / | | / ________ \ | | x / x | | |E *\/ E + 2 | | |-------------- + 3| dx | | x | | \ E / | / 0
$$\int\limits_{0}^{\pi} \left(3 + \frac{e^{x} \sqrt{e^{x} + 2}}{e^{x}}\right)\, dx$$
Integral((E^x*sqrt(E^x + 2))/E^x + 3, (x, 0, pi))
Respuesta (Indefinida)
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// / ________\ \
|| | ___ / x | |
/ || ___ |\/ 2 *\/ 2 + E | |
| ||-\/ 2 *acoth|-----------------| |
| / ________ \ || \ 2 / x |
| | x / x | ________ ||-------------------------------- for 2 + E > 2|
| |E *\/ E + 2 | / x || 2 |
| |-------------- + 3| dx = C + 2*\/ 2 + E + 3*x + 4*|< |
| | x | || / ________\ |
| \ E / || | ___ / x | |
| || ___ |\/ 2 *\/ 2 + E | |
/ ||-\/ 2 *atanh|-----------------| |
|| \ 2 / x |
||-------------------------------- for 2 + E < 2|
\\ 2 /
$$\int \left(3 + \frac{e^{x} \sqrt{e^{x} + 2}}{e^{x}}\right)\, dx = C + 3 x + 2 \sqrt{e^{x} + 2} + 4 \left(\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} \sqrt{e^{x} + 2}}{2} \right)}}{2} & \text{for}\: e^{x} + 2 > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} \sqrt{e^{x} + 2}}{2} \right)}}{2} & \text{for}\: e^{x} + 2 < 2 \end{cases}\right)$$
Respuesta
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/ _________\
_________ | ___ / pi | / ___\
___ / pi ___ |\/ 2 *\/ 2 + e | ___ |\/ 6 |
- 2*\/ 3 + 2*\/ 2 + e + 3*pi - 2*\/ 2 *acoth|------------------| + 2*\/ 2 *acoth|-----|
\ 2 / \ 2 /
$$- 2 \sqrt{3} - 2 \sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} \sqrt{2 + e^{\pi}}}{2} \right)} + 2 \sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{6}}{2} \right)} + 3 \pi + 2 \sqrt{2 + e^{\pi}}$$
=
=
/ _________\
_________ | ___ / pi | / ___\
___ / pi ___ |\/ 2 *\/ 2 + e | ___ |\/ 6 |
- 2*\/ 3 + 2*\/ 2 + e + 3*pi - 2*\/ 2 *acoth|------------------| + 2*\/ 2 *acoth|-----|
\ 2 / \ 2 /
$$- 2 \sqrt{3} - 2 \sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} \sqrt{2 + e^{\pi}}}{2} \right)} + 2 \sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{6}}{2} \right)} + 3 \pi + 2 \sqrt{2 + e^{\pi}}$$
-2*sqrt(3) + 2*sqrt(2 + exp(pi)) + 3*pi - 2*sqrt(2)*acoth(sqrt(2)*sqrt(2 + exp(pi))/2) + 2*sqrt(2)*acoth(sqrt(6)/2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.