Sr Examen
Integral de 1/(2+sin(x))^2 dx
Solución
Ha introducido
[src]
2*pi / | | 1 | ------------- dx | 2 | (2 + sin(x)) | / 0
$$\int\limits_{0}^{2 \pi} \frac{1}{\left(\sin{\left(x \right)} + 2\right)^{2}}\, dx$$
Integral(1/((2 + sin(x))^2), (x, 0, 2*pi))
Respuesta (Indefinida)
[src]
/ /x pi\ / ___ /x\\\ / /x pi\ / ___ /x\\\ / /x pi\ / ___ /x\\\
| |- - --| | ___ 2*\/ 3 *tan|-||| | |- - --| | ___ 2*\/ 3 *tan|-||| | |- - --| | ___ 2*\/ 3 *tan|-|||
/ /x\ ___ | |2 2 | |\/ 3 \2/|| ___ 2/x\ | |2 2 | |\/ 3 \2/|| ___ | |2 2 | |\/ 3 \2/|| /x\
| 3*tan|-| 8*\/ 3 *|pi*floor|------| + atan|----- + --------------|| 8*\/ 3 *tan |-|*|pi*floor|------| + atan|----- + --------------|| 8*\/ 3 *|pi*floor|------| + atan|----- + --------------||*tan|-|
| 1 6 \2/ \ \ pi / \ 3 3 // \2/ \ \ pi / \ 3 3 // \ \ pi / \ 3 3 // \2/
| ------------- dx = C + --------------------------- + --------------------------- + --------------------------------------------------------- + ----------------------------------------------------------------- + ----------------------------------------------------------------
| 2 2/x\ /x\ 2/x\ /x\ 2/x\ /x\ 2/x\ /x\ 2/x\ /x\
| (2 + sin(x)) 18 + 18*tan |-| + 18*tan|-| 18 + 18*tan |-| + 18*tan|-| 18 + 18*tan |-| + 18*tan|-| 18 + 18*tan |-| + 18*tan|-| 18 + 18*tan |-| + 18*tan|-|
| \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
/
$$\int \frac{1}{\left(\sin{\left(x \right)} + 2\right)^{2}}\, dx = C + \frac{8 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} + \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan^{2}{\left(\frac{x}{2} \right)}}{18 \tan^{2}{\left(\frac{x}{2} \right)} + 18 \tan{\left(\frac{x}{2} \right)} + 18} + \frac{8 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} + \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan{\left(\frac{x}{2} \right)}}{18 \tan^{2}{\left(\frac{x}{2} \right)} + 18 \tan{\left(\frac{x}{2} \right)} + 18} + \frac{8 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} + \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{18 \tan^{2}{\left(\frac{x}{2} \right)} + 18 \tan{\left(\frac{x}{2} \right)} + 18} + \frac{3 \tan{\left(\frac{x}{2} \right)}}{18 \tan^{2}{\left(\frac{x}{2} \right)} + 18 \tan{\left(\frac{x}{2} \right)} + 18} + \frac{6}{18 \tan^{2}{\left(\frac{x}{2} \right)} + 18 \tan{\left(\frac{x}{2} \right)} + 18}$$
Gráfica
Respuesta
[src]
___
4*pi*\/ 3
----------
9
$$\frac{4 \sqrt{3} \pi}{9}$$
=
=
___
4*pi*\/ 3
----------
9
$$\frac{4 \sqrt{3} \pi}{9}$$
4*pi*sqrt(3)/9
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.