Sr Examen
Integral de 1/(1+4sin^(2)x) dx
Solución
Ha introducido
[src]
pi -- 4 / | | 1 | ------------- dx | 2 | 1 + 4*sin (x) | / 0
$$\int\limits_{0}^{\frac{\pi}{4}} \frac{1}{4 \sin^{2}{\left(x \right)} + 1}\, dx$$
Integral(1/(1 + 4*sin(x)^2), (x, 0, pi/4))
Respuesta (Indefinida)
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/ /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/ ||
9*|pi*floor|------| + atan|----------------|| 4*\/ 5 *|pi*floor|------| + atan|----------------|| 9*\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 *|pi*floor|------| + atan|----------------|| 4*\/ 5 *\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 *|pi*floor|------| + atan|----------------||
/ | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________||
| | | / ___ || | | / ___ || | | / ___ || | | / ___ ||
| 1 \ \\/ 9 - 4*\/ 5 // \ \\/ 9 - 4*\/ 5 // \ \\/ 9 + 4*\/ 5 // \ \\/ 9 + 4*\/ 5 //
| ------------- dx = C + ----------------------------------------------- + --------------------------------------------------- + ------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
| 2 _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
| 1 + 4*sin (x) / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
| 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5
/
$$\int \frac{1}{4 \sin^{2}{\left(x \right)} + 1}\, dx = C + \frac{4 \sqrt{5} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{9 - 4 \sqrt{5}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{9 \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{9 - 4 \sqrt{5}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{4 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{4 \sqrt{5} + 9}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{9 \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{4 \sqrt{5} + 9}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}}$$
Gráfica
Respuesta
[src]
/ / ___ \\ / / ___ \\ _____________ _____________ / / ___ \\ _____________ _____________ / / ___ \\
| | -1 + \/ 2 || ___ | | -1 + \/ 2 || / ___ / ___ | | -1 + \/ 2 || ___ / ___ / ___ | | -1 + \/ 2 ||
9*|-pi + atan|----------------|| 4*\/ 5 *|-pi + atan|----------------|| 9*\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 *|-pi + atan|----------------|| 4*\/ 5 *\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 *|-pi + atan|----------------||
| | _____________|| | | _____________|| _____________ _____________ | | _____________|| _____________ _____________ | | _____________||
| | / ___ || ___ | | / ___ || / ___ / ___ | | / ___ || ___ / ___ / ___ | | / ___ ||
9*pi \ \\/ 9 - 4*\/ 5 // 4*pi*\/ 5 \ \\/ 9 - 4*\/ 5 // 9*pi*\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 \ \\/ 9 + 4*\/ 5 // 4*pi*\/ 5 *\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 \ \\/ 9 + 4*\/ 5 //
----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ------------------------------------------------------------------ + ----------------------------------------------- + ------------------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5
$$\frac{9 \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{4 \sqrt{5} + 9}} \right)}\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{4 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{4 \sqrt{5} + 9}} \right)}\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{9 \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{9 - 4 \sqrt{5}}} \right)}\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{4 \sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{9 - 4 \sqrt{5}}} \right)}\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{4 \sqrt{5} \pi}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{4 \sqrt{5} \pi \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9}}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{9 \pi}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{9 \pi \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9}}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}}$$
=
=
/ / ___ \\ / / ___ \\ _____________ _____________ / / ___ \\ _____________ _____________ / / ___ \\
| | -1 + \/ 2 || ___ | | -1 + \/ 2 || / ___ / ___ | | -1 + \/ 2 || ___ / ___ / ___ | | -1 + \/ 2 ||
9*|-pi + atan|----------------|| 4*\/ 5 *|-pi + atan|----------------|| 9*\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 *|-pi + atan|----------------|| 4*\/ 5 *\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 *|-pi + atan|----------------||
| | _____________|| | | _____________|| _____________ _____________ | | _____________|| _____________ _____________ | | _____________||
| | / ___ || ___ | | / ___ || / ___ / ___ | | / ___ || ___ / ___ / ___ | | / ___ ||
9*pi \ \\/ 9 - 4*\/ 5 // 4*pi*\/ 5 \ \\/ 9 - 4*\/ 5 // 9*pi*\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 \ \\/ 9 + 4*\/ 5 // 4*pi*\/ 5 *\/ 9 - 4*\/ 5 *\/ 9 + 4*\/ 5 \ \\/ 9 + 4*\/ 5 //
----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ------------------------------------------------------------------ + ----------------------------------------------- + ------------------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5 85*\/ 9 - 4*\/ 5 + 38*\/ 5 *\/ 9 - 4*\/ 5
$$\frac{9 \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{4 \sqrt{5} + 9}} \right)}\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{4 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{4 \sqrt{5} + 9}} \right)}\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{9 \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{9 - 4 \sqrt{5}}} \right)}\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{4 \sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{9 - 4 \sqrt{5}}} \right)}\right)}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{4 \sqrt{5} \pi}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{4 \sqrt{5} \pi \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9}}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{9 \pi}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}} + \frac{9 \pi \sqrt{9 - 4 \sqrt{5}} \sqrt{4 \sqrt{5} + 9}}{38 \sqrt{5} \sqrt{9 - 4 \sqrt{5}} + 85 \sqrt{9 - 4 \sqrt{5}}}$$
9*pi/(85*sqrt(9 - 4*sqrt(5)) + 38*sqrt(5)*sqrt(9 - 4*sqrt(5))) + 9*(-pi + atan((-1 + sqrt(2))/sqrt(9 - 4*sqrt(5))))/(85*sqrt(9 - 4*sqrt(5)) + 38*sqrt(5)*sqrt(9 - 4*sqrt(5))) + 4*pi*sqrt(5)/(85*sqrt(9 - 4*sqrt(5)) + 38*sqrt(5)*sqrt(9 - 4*sqrt(5))) + 4*sqrt(5)*(-pi + atan((-1 + sqrt(2))/sqrt(9 - 4*sqrt(5))))/(85*sqrt(9 - 4*sqrt(5)) + 38*sqrt(5)*sqrt(9 - 4*sqrt(5))) + 9*pi*sqrt(9 - 4*sqrt(5))*sqrt(9 + 4*sqrt(5))/(85*sqrt(9 - 4*sqrt(5)) + 38*sqrt(5)*sqrt(9 - 4*sqrt(5))) + 9*sqrt(9 - 4*sqrt(5))*sqrt(9 + 4*sqrt(5))*(-pi + atan((-1 + sqrt(2))/sqrt(9 + 4*sqrt(5))))/(85*sqrt(9 - 4*sqrt(5)) + 38*sqrt(5)*sqrt(9 - 4*sqrt(5))) + 4*pi*sqrt(5)*sqrt(9 - 4*sqrt(5))*sqrt(9 + 4*sqrt(5))/(85*sqrt(9 - 4*sqrt(5)) + 38*sqrt(5)*sqrt(9 - 4*sqrt(5))) + 4*sqrt(5)*sqrt(9 - 4*sqrt(5))*sqrt(9 + 4*sqrt(5))*(-pi + atan((-1 + sqrt(2))/sqrt(9 + 4*sqrt(5))))/(85*sqrt(9 - 4*sqrt(5)) + 38*sqrt(5)*sqrt(9 - 4*sqrt(5)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.