Sr Examen
Integral de 1/(3cosx+4)^2 dx
Solución
Ha introducido
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1 / | | 1 | --------------- dx | 2 | (3*cos(x) + 4) | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(3 \cos{\left(x \right)} + 4\right)^{2}}\, dx$$
Integral(1/((3*cos(x) + 4)^2), (x, 0, 1))
Respuesta (Indefinida)
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/ /x pi\ / ___ /x\\\ / /x pi\ / ___ /x\\\
| |- - --| |\/ 7 *tan|-||| | |- - --| |\/ 7 *tan|-|||
/ /x\ ___ | |2 2 | | \2/|| ___ 2/x\ | |2 2 | | \2/||
| 42*tan|-| 56*\/ 7 *|pi*floor|------| + atan|------------|| 8*\/ 7 *tan |-|*|pi*floor|------| + atan|------------||
| 1 \2/ \ \ pi / \ 7 // \2/ \ \ pi / \ 7 //
| --------------- dx = C - ---------------- + ------------------------------------------------ + -------------------------------------------------------
| 2 2/x\ 2/x\ 2/x\
| (3*cos(x) + 4) 343 + 49*tan |-| 343 + 49*tan |-| 343 + 49*tan |-|
| \2/ \2/ \2/
/
$$\int \frac{1}{\left(3 \cos{\left(x \right)} + 4\right)^{2}}\, dx = C + \frac{8 \sqrt{7} \left(\operatorname{atan}{\left(\frac{\sqrt{7} \tan{\left(\frac{x}{2} \right)}}{7} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan^{2}{\left(\frac{x}{2} \right)}}{49 \tan^{2}{\left(\frac{x}{2} \right)} + 343} + \frac{56 \sqrt{7} \left(\operatorname{atan}{\left(\frac{\sqrt{7} \tan{\left(\frac{x}{2} \right)}}{7} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{49 \tan^{2}{\left(\frac{x}{2} \right)} + 343} - \frac{42 \tan{\left(\frac{x}{2} \right)}}{49 \tan^{2}{\left(\frac{x}{2} \right)} + 343}$$
Gráfica
Respuesta
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/ / ___ \\ / / ___ \\
___ | |\/ 7 *tan(1/2)|| ___ 2 | |\/ 7 *tan(1/2)||
___ 56*\/ 7 *|-pi + atan|--------------|| 8*\/ 7 *tan (1/2)*|-pi + atan|--------------||
42*tan(1/2) 8*pi*\/ 7 \ \ 7 // \ \ 7 //
- ------------------ + ---------- + ------------------------------------- + ----------------------------------------------
2 49 2 2
343 + 49*tan (1/2) 343 + 49*tan (1/2) 343 + 49*tan (1/2)
$$\frac{56 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{7} \tan{\left(\frac{1}{2} \right)}}{7} \right)}\right)}{49 \tan^{2}{\left(\frac{1}{2} \right)} + 343} - \frac{42 \tan{\left(\frac{1}{2} \right)}}{49 \tan^{2}{\left(\frac{1}{2} \right)} + 343} + \frac{8 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{7} \tan{\left(\frac{1}{2} \right)}}{7} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{49 \tan^{2}{\left(\frac{1}{2} \right)} + 343} + \frac{8 \sqrt{7} \pi}{49}$$
=
=
/ / ___ \\ / / ___ \\
___ | |\/ 7 *tan(1/2)|| ___ 2 | |\/ 7 *tan(1/2)||
___ 56*\/ 7 *|-pi + atan|--------------|| 8*\/ 7 *tan (1/2)*|-pi + atan|--------------||
42*tan(1/2) 8*pi*\/ 7 \ \ 7 // \ \ 7 //
- ------------------ + ---------- + ------------------------------------- + ----------------------------------------------
2 49 2 2
343 + 49*tan (1/2) 343 + 49*tan (1/2) 343 + 49*tan (1/2)
$$\frac{56 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{7} \tan{\left(\frac{1}{2} \right)}}{7} \right)}\right)}{49 \tan^{2}{\left(\frac{1}{2} \right)} + 343} - \frac{42 \tan{\left(\frac{1}{2} \right)}}{49 \tan^{2}{\left(\frac{1}{2} \right)} + 343} + \frac{8 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{7} \tan{\left(\frac{1}{2} \right)}}{7} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{49 \tan^{2}{\left(\frac{1}{2} \right)} + 343} + \frac{8 \sqrt{7} \pi}{49}$$
-42*tan(1/2)/(343 + 49*tan(1/2)^2) + 8*pi*sqrt(7)/49 + 56*sqrt(7)*(-pi + atan(sqrt(7)*tan(1/2)/7))/(343 + 49*tan(1/2)^2) + 8*sqrt(7)*tan(1/2)^2*(-pi + atan(sqrt(7)*tan(1/2)/7))/(343 + 49*tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.