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