Sr Examen
Integral de 1/(1+8*sinx^2) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | 2 | 1 + 8*sin (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{8 \sin^{2}{\left(x \right)} + 1}\, dx$$
Integral(1/(1 + 8*sin(x)^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
/x pi\ / /x\ \ / /x pi\ / /x\ \\
|- - --| | tan|-| | _______________ _______________ | |- - --| | tan|-| ||
|2 2 | | \2/ | / ___ / ___ | |2 2 | | \2/ ||
pi*floor|------| + atan|------------------| \/ 17 - 12*\/ 2 *\/ 17 + 12*\/ 2 *|pi*floor|------| + atan|------------------||
/ \ pi / | _______________| | \ pi / | _______________||
| | / ___ | | | / ___ ||
| 1 \\/ 17 - 12*\/ 2 / \ \\/ 17 + 12*\/ 2 //
| ------------- dx = C + ------------------------------------------------- + -----------------------------------------------------------------------------------
| 2 _______________ _______________ _______________ _______________
| 1 + 8*sin (x) / ___ ___ / ___ / ___ ___ / ___
| 9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2 9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2
/
$$\int \frac{1}{8 \sin^{2}{\left(x \right)} + 1}\, dx = C + \frac{\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{17 - 12 \sqrt{2}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}} + \frac{\sqrt{17 - 12 \sqrt{2}} \sqrt{12 \sqrt{2} + 17} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{12 \sqrt{2} + 17}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}}$$
Gráfica
Respuesta
[src]
_______________ _______________
/ tan(1/2) \ / ___ / ___ / / tan(1/2) \\
-pi + atan|------------------| \/ 17 - 12*\/ 2 *\/ 17 + 12*\/ 2 *|-pi + atan|------------------||
| _______________| _______________ _______________ | | _______________||
| / ___ | / ___ / ___ | | / ___ ||
pi \\/ 17 - 12*\/ 2 / pi*\/ 17 - 12*\/ 2 *\/ 17 + 12*\/ 2 \ \\/ 17 + 12*\/ 2 //
------------------------------------------------- + ------------------------------------------------- + ------------------------------------------------- + ----------------------------------------------------------------------
_______________ _______________ _______________ _______________ _______________ _______________ _______________ _______________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2 9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2 9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2 9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2
$$\frac{\sqrt{17 - 12 \sqrt{2}} \sqrt{12 \sqrt{2} + 17} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{12 \sqrt{2} + 17}} \right)}\right)}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{17 - 12 \sqrt{2}}} \right)}}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}} + \frac{\pi}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}} + \frac{\pi \sqrt{17 - 12 \sqrt{2}} \sqrt{12 \sqrt{2} + 17}}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}}$$
=
=
_______________ _______________
/ tan(1/2) \ / ___ / ___ / / tan(1/2) \\
-pi + atan|------------------| \/ 17 - 12*\/ 2 *\/ 17 + 12*\/ 2 *|-pi + atan|------------------||
| _______________| _______________ _______________ | | _______________||
| / ___ | / ___ / ___ | | / ___ ||
pi \\/ 17 - 12*\/ 2 / pi*\/ 17 - 12*\/ 2 *\/ 17 + 12*\/ 2 \ \\/ 17 + 12*\/ 2 //
------------------------------------------------- + ------------------------------------------------- + ------------------------------------------------- + ----------------------------------------------------------------------
_______________ _______________ _______________ _______________ _______________ _______________ _______________ _______________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2 9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2 9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2 9*\/ 17 - 12*\/ 2 + 6*\/ 2 *\/ 17 - 12*\/ 2
$$\frac{\sqrt{17 - 12 \sqrt{2}} \sqrt{12 \sqrt{2} + 17} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{12 \sqrt{2} + 17}} \right)}\right)}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{17 - 12 \sqrt{2}}} \right)}}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}} + \frac{\pi}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}} + \frac{\pi \sqrt{17 - 12 \sqrt{2}} \sqrt{12 \sqrt{2} + 17}}{6 \sqrt{2} \sqrt{17 - 12 \sqrt{2}} + 9 \sqrt{17 - 12 \sqrt{2}}}$$
pi/(9*sqrt(17 - 12*sqrt(2)) + 6*sqrt(2)*sqrt(17 - 12*sqrt(2))) + (-pi + atan(tan(1/2)/sqrt(17 - 12*sqrt(2))))/(9*sqrt(17 - 12*sqrt(2)) + 6*sqrt(2)*sqrt(17 - 12*sqrt(2))) + pi*sqrt(17 - 12*sqrt(2))*sqrt(17 + 12*sqrt(2))/(9*sqrt(17 - 12*sqrt(2)) + 6*sqrt(2)*sqrt(17 - 12*sqrt(2))) + sqrt(17 - 12*sqrt(2))*sqrt(17 + 12*sqrt(2))*(-pi + atan(tan(1/2)/sqrt(17 + 12*sqrt(2))))/(9*sqrt(17 - 12*sqrt(2)) + 6*sqrt(2)*sqrt(17 - 12*sqrt(2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.