Sr Examen
Integral de 1/(1+sin²x) dx
Solución
Ha introducido
[src]
1 / | | 1 | ----------- dx | 2 | 1 + sin (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sin^{2}{\left(x \right)} + 1}\, dx$$
Integral(1/(1 + sin(x)^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ /x pi\ / /x\ \\ / /x pi\ / /x\ \\ / /x pi\ / /x\ \\ / /x pi\ / /x\ \\
| |- - --| | tan|-| || | |- - --| | tan|-| || _____________ _____________ | |- - --| | tan|-| || _____________ _____________ | |- - --| | tan|-| ||
| |2 2 | | \2/ || ___ | |2 2 | | \2/ || / ___ / ___ | |2 2 | | \2/ || ___ / ___ / ___ | |2 2 | | \2/ ||
3*|pi*floor|------| + atan|----------------|| 2*\/ 2 *|pi*floor|------| + atan|----------------|| 3*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *|pi*floor|------| + atan|----------------|| 2*\/ 2 *\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *|pi*floor|------| + atan|----------------||
/ | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________||
| | | / ___ || | | / ___ || | | / ___ || | | / ___ ||
| 1 \ \\/ 3 - 2*\/ 2 // \ \\/ 3 - 2*\/ 2 // \ \\/ 3 + 2*\/ 2 // \ \\/ 3 + 2*\/ 2 //
| ----------- dx = C + ---------------------------------------------- + --------------------------------------------------- + ------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
| 2 _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
| 1 + sin (x) / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
| 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2
/
$$\int \frac{1}{\sin^{2}{\left(x \right)} + 1}\, dx = C + \frac{2 \sqrt{2} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}}$$
Gráfica
Respuesta
[src]
_____________ _____________
/ tan(1/2) \ / ___ / ___ / / tan(1/2) \\
-pi + atan|----------------| \/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *|-pi + atan|----------------||
| _____________| _____________ _____________ | | _____________||
| / ___ | / ___ / ___ | | / ___ ||
pi \\/ 3 - 2*\/ 2 / pi*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 \ \\/ 3 + 2*\/ 2 //
------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ----------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2
$$\frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}\right)}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{\pi}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{\pi \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}}$$
=
=
_____________ _____________
/ tan(1/2) \ / ___ / ___ / / tan(1/2) \\
-pi + atan|----------------| \/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *|-pi + atan|----------------||
| _____________| _____________ _____________ | | _____________||
| / ___ | / ___ / ___ | | / ___ ||
pi \\/ 3 - 2*\/ 2 / pi*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 \ \\/ 3 + 2*\/ 2 //
------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ----------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2
$$\frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}\right)}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{\pi}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{\pi \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}}$$
pi/(2*sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2))) + (-pi + atan(tan(1/2)/sqrt(3 - 2*sqrt(2))))/(2*sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2))) + pi*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))/(2*sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2))) + sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*(-pi + atan(tan(1/2)/sqrt(3 + 2*sqrt(2))))/(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.