Sr Examen
Integral de 1/(3+2*sin(x)^2) dx
Solución
Ha introducido
[src]
2*pi / | | 1 | ------------- dx | 2 | 3 + 2*sin (x) | / 0
$$\int\limits_{0}^{2 \pi} \frac{1}{2 \sin^{2}{\left(x \right)} + 3}\, dx$$
Integral(1/(3 + 2*sin(x)^2), (x, 0, 2*pi))
Respuesta (Indefinida)
[src]
/ /x pi\ / ___ /x\ \\ / /x pi\ / ___ /x\ \\ / /x pi\ / ___ /x\ \\ / /x pi\ / ___ /x\ \\
| |- - --| | \/ 3 *tan|-| || | |- - --| | \/ 3 *tan|-| || ______________ ______________ | |- - --| | \/ 3 *tan|-| || ______________ ______________ | |- - --| | \/ 3 *tan|-| ||
____ | |2 2 | | \2/ || ___ | |2 2 | | \2/ || ____ / ____ / ____ | |2 2 | | \2/ || ___ / ____ / ____ | |2 2 | | \2/ ||
6*\/ 30 *|pi*floor|------| + atan|-----------------|| 21*\/ 3 *|pi*floor|------| + atan|-----------------|| 2*\/ 30 *\/ 7 - 2*\/ 10 *\/ 7 + 2*\/ 10 *|pi*floor|------| + atan|-----------------|| 7*\/ 3 *\/ 7 - 2*\/ 10 *\/ 7 + 2*\/ 10 *|pi*floor|------| + atan|-----------------||
/ | \ pi / | ______________|| | \ pi / | ______________|| | \ pi / | ______________|| | \ pi / | ______________||
| | | / ____ || | | / ____ || | | / ____ || | | / ____ ||
| 1 \ \\/ 7 - 2*\/ 10 // \ \\/ 7 - 2*\/ 10 // \ \\/ 7 + 2*\/ 10 // \ \\/ 7 + 2*\/ 10 //
| ------------- dx = C + ----------------------------------------------------- + ----------------------------------------------------- + ----------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------
| 2 ______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________
| 3 + 2*sin (x) / ____ ____ / ____ / ____ ____ / ____ / ____ ____ / ____ / ____ ____ / ____
| 165*\/ 7 - 2*\/ 10 + 51*\/ 10 *\/ 7 - 2*\/ 10 165*\/ 7 - 2*\/ 10 + 51*\/ 10 *\/ 7 - 2*\/ 10 165*\/ 7 - 2*\/ 10 + 51*\/ 10 *\/ 7 - 2*\/ 10 165*\/ 7 - 2*\/ 10 + 51*\/ 10 *\/ 7 - 2*\/ 10
/
$$\int \frac{1}{2 \sin^{2}{\left(x \right)} + 3}\, dx = C + \frac{6 \sqrt{30} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{\sqrt{7 - 2 \sqrt{10}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{51 \sqrt{10} \sqrt{7 - 2 \sqrt{10}} + 165 \sqrt{7 - 2 \sqrt{10}}} + \frac{21 \sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{\sqrt{7 - 2 \sqrt{10}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{51 \sqrt{10} \sqrt{7 - 2 \sqrt{10}} + 165 \sqrt{7 - 2 \sqrt{10}}} + \frac{2 \sqrt{30} \sqrt{7 - 2 \sqrt{10}} \sqrt{2 \sqrt{10} + 7} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{10} + 7}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{51 \sqrt{10} \sqrt{7 - 2 \sqrt{10}} + 165 \sqrt{7 - 2 \sqrt{10}}} + \frac{7 \sqrt{3} \sqrt{7 - 2 \sqrt{10}} \sqrt{2 \sqrt{10} + 7} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{10} + 7}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{51 \sqrt{10} \sqrt{7 - 2 \sqrt{10}} + 165 \sqrt{7 - 2 \sqrt{10}}}$$
Gráfica
Respuesta
[src]
______________ ______________
___ ___ / ____ / ____
3*pi*\/ 3 pi*\/ 3 *\/ 7 - 2*\/ 10 *\/ 7 + 2*\/ 10
------------------------------------------------- + -------------------------------------------------
______________ ______________ ______________ ______________
/ ____ ____ / ____ / ____ ____ / ____
15*\/ 7 - 2*\/ 10 + 3*\/ 10 *\/ 7 - 2*\/ 10 15*\/ 7 - 2*\/ 10 + 3*\/ 10 *\/ 7 - 2*\/ 10
$$\frac{3 \sqrt{3} \pi}{3 \sqrt{10} \sqrt{7 - 2 \sqrt{10}} + 15 \sqrt{7 - 2 \sqrt{10}}} + \frac{\sqrt{3} \pi \sqrt{7 - 2 \sqrt{10}} \sqrt{2 \sqrt{10} + 7}}{3 \sqrt{10} \sqrt{7 - 2 \sqrt{10}} + 15 \sqrt{7 - 2 \sqrt{10}}}$$
=
=
______________ ______________
___ ___ / ____ / ____
3*pi*\/ 3 pi*\/ 3 *\/ 7 - 2*\/ 10 *\/ 7 + 2*\/ 10
------------------------------------------------- + -------------------------------------------------
______________ ______________ ______________ ______________
/ ____ ____ / ____ / ____ ____ / ____
15*\/ 7 - 2*\/ 10 + 3*\/ 10 *\/ 7 - 2*\/ 10 15*\/ 7 - 2*\/ 10 + 3*\/ 10 *\/ 7 - 2*\/ 10
$$\frac{3 \sqrt{3} \pi}{3 \sqrt{10} \sqrt{7 - 2 \sqrt{10}} + 15 \sqrt{7 - 2 \sqrt{10}}} + \frac{\sqrt{3} \pi \sqrt{7 - 2 \sqrt{10}} \sqrt{2 \sqrt{10} + 7}}{3 \sqrt{10} \sqrt{7 - 2 \sqrt{10}} + 15 \sqrt{7 - 2 \sqrt{10}}}$$
3*pi*sqrt(3)/(15*sqrt(7 - 2*sqrt(10)) + 3*sqrt(10)*sqrt(7 - 2*sqrt(10))) + pi*sqrt(3)*sqrt(7 - 2*sqrt(10))*sqrt(7 + 2*sqrt(10))/(15*sqrt(7 - 2*sqrt(10)) + 3*sqrt(10)*sqrt(7 - 2*sqrt(10)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.