Sr Examen
Integral de 1/(5+4cosx)^2 dx
Solución
Ha introducido
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1 / | | 1 | --------------- dx | 2 | (5 + 4*cos(x)) | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(4 \cos{\left(x \right)} + 5\right)^{2}}\, dx$$
Integral(1/((5 + 4*cos(x))^2), (x, 0, 1))
Respuesta (Indefinida)
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/ /x pi\ / /x\\\ / /x pi\ / /x\\\
| |- - --| |tan|-||| | |- - --| |tan|-|||
/ /x\ | |2 2 | | \2/|| 2/x\ | |2 2 | | \2/||
| 24*tan|-| 90*|pi*floor|------| + atan|------|| 10*tan |-|*|pi*floor|------| + atan|------||
| 1 \2/ \ \ pi / \ 3 // \2/ \ \ pi / \ 3 //
| --------------- dx = C - ---------------- + ------------------------------------ + --------------------------------------------
| 2 2/x\ 2/x\ 2/x\
| (5 + 4*cos(x)) 243 + 27*tan |-| 243 + 27*tan |-| 243 + 27*tan |-|
| \2/ \2/ \2/
/
$$\int \frac{1}{\left(4 \cos{\left(x \right)} + 5\right)^{2}}\, dx = C + \frac{10 \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan^{2}{\left(\frac{x}{2} \right)}}{27 \tan^{2}{\left(\frac{x}{2} \right)} + 243} + \frac{90 \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{27 \tan^{2}{\left(\frac{x}{2} \right)} + 243} - \frac{24 \tan{\left(\frac{x}{2} \right)}}{27 \tan^{2}{\left(\frac{x}{2} \right)} + 243}$$
Gráfica
Respuesta
[src]
/ /tan(1/2)\\ 2 / /tan(1/2)\\
90*|-pi + atan|--------|| 10*tan (1/2)*|-pi + atan|--------||
10*pi 24*tan(1/2) \ \ 3 // \ \ 3 //
----- - ------------------ + ------------------------- + -----------------------------------
27 2 2 2
243 + 27*tan (1/2) 243 + 27*tan (1/2) 243 + 27*tan (1/2)
$$\frac{90 \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{27 \tan^{2}{\left(\frac{1}{2} \right)} + 243} - \frac{24 \tan{\left(\frac{1}{2} \right)}}{27 \tan^{2}{\left(\frac{1}{2} \right)} + 243} + \frac{10 \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{27 \tan^{2}{\left(\frac{1}{2} \right)} + 243} + \frac{10 \pi}{27}$$
=
=
/ /tan(1/2)\\ 2 / /tan(1/2)\\
90*|-pi + atan|--------|| 10*tan (1/2)*|-pi + atan|--------||
10*pi 24*tan(1/2) \ \ 3 // \ \ 3 //
----- - ------------------ + ------------------------- + -----------------------------------
27 2 2 2
243 + 27*tan (1/2) 243 + 27*tan (1/2) 243 + 27*tan (1/2)
$$\frac{90 \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{27 \tan^{2}{\left(\frac{1}{2} \right)} + 243} - \frac{24 \tan{\left(\frac{1}{2} \right)}}{27 \tan^{2}{\left(\frac{1}{2} \right)} + 243} + \frac{10 \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{3} \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{27 \tan^{2}{\left(\frac{1}{2} \right)} + 243} + \frac{10 \pi}{27}$$
10*pi/27 - 24*tan(1/2)/(243 + 27*tan(1/2)^2) + 90*(-pi + atan(tan(1/2)/3))/(243 + 27*tan(1/2)^2) + 10*tan(1/2)^2*(-pi + atan(tan(1/2)/3))/(243 + 27*tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.