Sr Examen
Integral de 1/((chx)^2+(shx)^2) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------------- dx | 2 2 | cosh (x) + sinh (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sinh^{2}{\left(x \right)} + \cosh^{2}{\left(x \right)}}\, dx$$
Integral(1/(cosh(x)^2 + sinh(x)^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ /x\ \ / /x\ \
| tanh|-| | _____________ _____________ | tanh|-| |
| \2/ | / ___ / ___ | \2/ |
atan|----------------| \/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *atan|----------------|
/ | _____________| | _____________|
| | / ___ | | / ___ |
| 1 \\/ 3 - 2*\/ 2 / \\/ 3 + 2*\/ 2 /
| ------------------- dx = C + ----------------------------------------- - --------------------------------------------------------
| 2 2 _____________ _____________ _____________ _____________
| cosh (x) + sinh (x) / ___ ___ / ___ / ___ ___ / ___
| \/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 \/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2
/
$$\int \frac{1}{\sinh^{2}{\left(x \right)} + \cosh^{2}{\left(x \right)}}\, dx = C + \frac{\operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
Gráfica
Respuesta
[src]
_____________ _____________
/ tanh(1/2) \ / ___ / ___ / tanh(1/2) \
atan|----------------| \/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *atan|----------------|
| _____________| | _____________|
| / ___ | | / ___ |
\\/ 3 - 2*\/ 2 / \\/ 3 + 2*\/ 2 /
----------------------------------------- - --------------------------------------------------------
_____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___
\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 \/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2
$$- \frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{\operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
=
=
_____________ _____________
/ tanh(1/2) \ / ___ / ___ / tanh(1/2) \
atan|----------------| \/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *atan|----------------|
| _____________| | _____________|
| / ___ | | / ___ |
\\/ 3 - 2*\/ 2 / \\/ 3 + 2*\/ 2 /
----------------------------------------- - --------------------------------------------------------
_____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___
\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 \/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2
$$- \frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{\operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
atan(tanh(1/2)/sqrt(3 - 2*sqrt(2)))/(sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2))) - sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*atan(tanh(1/2)/sqrt(3 + 2*sqrt(2)))/(sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.