Sr Examen
Integral de 1/(2+sinx)^2 dx
Solución
Ha introducido
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1 / | | 1 | ------------- dx | 2 | (2 + sin(x)) | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(\sin{\left(x \right)} + 2\right)^{2}}\, dx$$
Integral(1/((2 + sin(x))^2), (x, 0, 1))
Respuesta (Indefinida)
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/ /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
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/ / ___ ___ \\ / / ___ ___ \\ / / ___ ___ \\
___ | |\/ 3 2*\/ 3 *tan(1/2)|| ___ 2 | |\/ 3 2*\/ 3 *tan(1/2)|| ___ | |\/ 3 2*\/ 3 *tan(1/2)||
___ 8*\/ 3 *|-pi + atan|----- + ----------------|| 8*\/ 3 *tan (1/2)*|-pi + atan|----- + ----------------|| 8*\/ 3 *|-pi + atan|----- + ----------------||*tan(1/2)
1 6 3*tan(1/2) 10*pi*\/ 3 \ \ 3 3 // \ \ 3 3 // \ \ 3 3 //
- - + ------------------------------- + ------------------------------- + ----------- + ---------------------------------------------- + -------------------------------------------------------- + -------------------------------------------------------
3 2 2 27 2 2 2
18 + 18*tan (1/2) + 18*tan(1/2) 18 + 18*tan (1/2) + 18*tan(1/2) 18 + 18*tan (1/2) + 18*tan(1/2) 18 + 18*tan (1/2) + 18*tan(1/2) 18 + 18*tan (1/2) + 18*tan(1/2)
$$\frac{8 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} + \frac{8 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan{\left(\frac{1}{2} \right)}}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} - \frac{1}{3} + \frac{8 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} + \frac{3 \tan{\left(\frac{1}{2} \right)}}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} + \frac{6}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} + \frac{10 \sqrt{3} \pi}{27}$$
=
=
/ / ___ ___ \\ / / ___ ___ \\ / / ___ ___ \\
___ | |\/ 3 2*\/ 3 *tan(1/2)|| ___ 2 | |\/ 3 2*\/ 3 *tan(1/2)|| ___ | |\/ 3 2*\/ 3 *tan(1/2)||
___ 8*\/ 3 *|-pi + atan|----- + ----------------|| 8*\/ 3 *tan (1/2)*|-pi + atan|----- + ----------------|| 8*\/ 3 *|-pi + atan|----- + ----------------||*tan(1/2)
1 6 3*tan(1/2) 10*pi*\/ 3 \ \ 3 3 // \ \ 3 3 // \ \ 3 3 //
- - + ------------------------------- + ------------------------------- + ----------- + ---------------------------------------------- + -------------------------------------------------------- + -------------------------------------------------------
3 2 2 27 2 2 2
18 + 18*tan (1/2) + 18*tan(1/2) 18 + 18*tan (1/2) + 18*tan(1/2) 18 + 18*tan (1/2) + 18*tan(1/2) 18 + 18*tan (1/2) + 18*tan(1/2) 18 + 18*tan (1/2) + 18*tan(1/2)
$$\frac{8 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} + \frac{8 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan{\left(\frac{1}{2} \right)}}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} - \frac{1}{3} + \frac{8 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} + \frac{3 \tan{\left(\frac{1}{2} \right)}}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} + \frac{6}{18 \tan^{2}{\left(\frac{1}{2} \right)} + 18 \tan{\left(\frac{1}{2} \right)} + 18} + \frac{10 \sqrt{3} \pi}{27}$$
-1/3 + 6/(18 + 18*tan(1/2)^2 + 18*tan(1/2)) + 3*tan(1/2)/(18 + 18*tan(1/2)^2 + 18*tan(1/2)) + 10*pi*sqrt(3)/27 + 8*sqrt(3)*(-pi + atan(sqrt(3)/3 + 2*sqrt(3)*tan(1/2)/3))/(18 + 18*tan(1/2)^2 + 18*tan(1/2)) + 8*sqrt(3)*tan(1/2)^2*(-pi + atan(sqrt(3)/3 + 2*sqrt(3)*tan(1/2)/3))/(18 + 18*tan(1/2)^2 + 18*tan(1/2)) + 8*sqrt(3)*(-pi + atan(sqrt(3)/3 + 2*sqrt(3)*tan(1/2)/3))*tan(1/2)/(18 + 18*tan(1/2)^2 + 18*tan(1/2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.