Sr Examen
Integral de 1/(2+cosx)^2 dx
Solución
Ha introducido
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1 / | | 1 | ------------- dx | 2 | (2 + cos(x)) | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(\cos{\left(x \right)} + 2\right)^{2}}\, dx$$
Integral(1/((2 + cos(x))^2), (x, 0, 1))
Respuesta (Indefinida)
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/ /x pi\ / ___ /x\\\ / /x pi\ / ___ /x\\\
| |- - --| |\/ 3 *tan|-||| | |- - --| |\/ 3 *tan|-|||
/ /x\ ___ | |2 2 | | \2/|| ___ 2/x\ | |2 2 | | \2/||
| 6*tan|-| 12*\/ 3 *|pi*floor|------| + atan|------------|| 4*\/ 3 *tan |-|*|pi*floor|------| + atan|------------||
| 1 \2/ \ \ pi / \ 3 // \2/ \ \ pi / \ 3 //
| ------------- dx = C - -------------- + ------------------------------------------------ + -------------------------------------------------------
| 2 2/x\ 2/x\ 2/x\
| (2 + cos(x)) 27 + 9*tan |-| 27 + 9*tan |-| 27 + 9*tan |-|
| \2/ \2/ \2/
/
$$\int \frac{1}{\left(\cos{\left(x \right)} + 2\right)^{2}}\, dx = C + \frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan^{2}{\left(\frac{x}{2} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 27} + \frac{12 \sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 27} - \frac{6 \tan{\left(\frac{x}{2} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 27}$$
Gráfica
Respuesta
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/ / ___ \\ / / ___ \\
___ | |\/ 3 *tan(1/2)|| ___ 2 | |\/ 3 *tan(1/2)||
___ 12*\/ 3 *|-pi + atan|--------------|| 4*\/ 3 *tan (1/2)*|-pi + atan|--------------||
6*tan(1/2) 4*pi*\/ 3 \ \ 3 // \ \ 3 //
- ---------------- + ---------- + ------------------------------------- + ----------------------------------------------
2 9 2 2
27 + 9*tan (1/2) 27 + 9*tan (1/2) 27 + 9*tan (1/2)
$$\frac{12 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} + \frac{4 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} - \frac{6 \tan{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} + \frac{4 \sqrt{3} \pi}{9}$$
=
=
/ / ___ \\ / / ___ \\
___ | |\/ 3 *tan(1/2)|| ___ 2 | |\/ 3 *tan(1/2)||
___ 12*\/ 3 *|-pi + atan|--------------|| 4*\/ 3 *tan (1/2)*|-pi + atan|--------------||
6*tan(1/2) 4*pi*\/ 3 \ \ 3 // \ \ 3 //
- ---------------- + ---------- + ------------------------------------- + ----------------------------------------------
2 9 2 2
27 + 9*tan (1/2) 27 + 9*tan (1/2) 27 + 9*tan (1/2)
$$\frac{12 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} + \frac{4 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} - \frac{6 \tan{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 27} + \frac{4 \sqrt{3} \pi}{9}$$
-6*tan(1/2)/(27 + 9*tan(1/2)^2) + 4*pi*sqrt(3)/9 + 12*sqrt(3)*(-pi + atan(sqrt(3)*tan(1/2)/3))/(27 + 9*tan(1/2)^2) + 4*sqrt(3)*tan(1/2)^2*(-pi + atan(sqrt(3)*tan(1/2)/3))/(27 + 9*tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.