Sr Examen
Integral de 1/(1+3sin^2x) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | 2 | 1 + 3*sin (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{3 \sin^{2}{\left(x \right)} + 1}\, dx$$
Integral(1/(1 + 3*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/ ||
7*|pi*floor|------| + atan|----------------|| 4*\/ 3 *|pi*floor|------| + atan|----------------|| 7*\/ 7 - 4*\/ 3 *\/ 7 + 4*\/ 3 *|pi*floor|------| + atan|----------------|| 4*\/ 3 *\/ 7 - 4*\/ 3 *\/ 7 + 4*\/ 3 *|pi*floor|------| + atan|----------------||
/ | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________||
| | | / ___ || | | / ___ || | | / ___ || | | / ___ ||
| 1 \ \\/ 7 - 4*\/ 3 // \ \\/ 7 - 4*\/ 3 // \ \\/ 7 + 4*\/ 3 // \ \\/ 7 + 4*\/ 3 //
| ------------- dx = C + ----------------------------------------------- + --------------------------------------------------- + ------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
| 2 _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
| 1 + 3*sin (x) / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
| 52*\/ 7 - 4*\/ 3 + 30*\/ 3 *\/ 7 - 4*\/ 3 52*\/ 7 - 4*\/ 3 + 30*\/ 3 *\/ 7 - 4*\/ 3 52*\/ 7 - 4*\/ 3 + 30*\/ 3 *\/ 7 - 4*\/ 3 52*\/ 7 - 4*\/ 3 + 30*\/ 3 *\/ 7 - 4*\/ 3
/
$$\int \frac{1}{3 \sin^{2}{\left(x \right)} + 1}\, dx = C + \frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{7 - 4 \sqrt{3}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{30 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 52 \sqrt{7 - 4 \sqrt{3}}} + \frac{7 \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{7 - 4 \sqrt{3}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{30 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 52 \sqrt{7 - 4 \sqrt{3}}} + \frac{4 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{4 \sqrt{3} + 7}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{30 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 52 \sqrt{7 - 4 \sqrt{3}}} + \frac{7 \sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{4 \sqrt{3} + 7}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{30 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 52 \sqrt{7 - 4 \sqrt{3}}}$$
Gráfica
Respuesta
[src]
_____________ _____________
/ tan(1/2) \ / ___ / ___ / / tan(1/2) \\
-pi + atan|----------------| \/ 7 - 4*\/ 3 *\/ 7 + 4*\/ 3 *|-pi + atan|----------------||
| _____________| _____________ _____________ | | _____________||
| / ___ | / ___ / ___ | | / ___ ||
pi \\/ 7 - 4*\/ 3 / pi*\/ 7 - 4*\/ 3 *\/ 7 + 4*\/ 3 \ \\/ 7 + 4*\/ 3 //
--------------------------------------------- + --------------------------------------------- + --------------------------------------------- + ----------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
4*\/ 7 - 4*\/ 3 + 2*\/ 3 *\/ 7 - 4*\/ 3 4*\/ 7 - 4*\/ 3 + 2*\/ 3 *\/ 7 - 4*\/ 3 4*\/ 7 - 4*\/ 3 + 2*\/ 3 *\/ 7 - 4*\/ 3 4*\/ 7 - 4*\/ 3 + 2*\/ 3 *\/ 7 - 4*\/ 3
$$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{4 \sqrt{3} + 7}} \right)}\right)}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{7 - 4 \sqrt{3}}} \right)}}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{\pi \sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7}}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{\pi}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}}$$
=
=
_____________ _____________
/ tan(1/2) \ / ___ / ___ / / tan(1/2) \\
-pi + atan|----------------| \/ 7 - 4*\/ 3 *\/ 7 + 4*\/ 3 *|-pi + atan|----------------||
| _____________| _____________ _____________ | | _____________||
| / ___ | / ___ / ___ | | / ___ ||
pi \\/ 7 - 4*\/ 3 / pi*\/ 7 - 4*\/ 3 *\/ 7 + 4*\/ 3 \ \\/ 7 + 4*\/ 3 //
--------------------------------------------- + --------------------------------------------- + --------------------------------------------- + ----------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
4*\/ 7 - 4*\/ 3 + 2*\/ 3 *\/ 7 - 4*\/ 3 4*\/ 7 - 4*\/ 3 + 2*\/ 3 *\/ 7 - 4*\/ 3 4*\/ 7 - 4*\/ 3 + 2*\/ 3 *\/ 7 - 4*\/ 3 4*\/ 7 - 4*\/ 3 + 2*\/ 3 *\/ 7 - 4*\/ 3
$$\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{4 \sqrt{3} + 7}} \right)}\right)}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{7 - 4 \sqrt{3}}} \right)}}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{\pi \sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7}}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{\pi}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}}$$
pi/(4*sqrt(7 - 4*sqrt(3)) + 2*sqrt(3)*sqrt(7 - 4*sqrt(3))) + (-pi + atan(tan(1/2)/sqrt(7 - 4*sqrt(3))))/(4*sqrt(7 - 4*sqrt(3)) + 2*sqrt(3)*sqrt(7 - 4*sqrt(3))) + pi*sqrt(7 - 4*sqrt(3))*sqrt(7 + 4*sqrt(3))/(4*sqrt(7 - 4*sqrt(3)) + 2*sqrt(3)*sqrt(7 - 4*sqrt(3))) + sqrt(7 - 4*sqrt(3))*sqrt(7 + 4*sqrt(3))*(-pi + atan(tan(1/2)/sqrt(7 + 4*sqrt(3))))/(4*sqrt(7 - 4*sqrt(3)) + 2*sqrt(3)*sqrt(7 - 4*sqrt(3)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.