Sr Examen
Integral de x/(cosx)^2 dx
Solución
Ha introducido
[src]
pi -- 4 / | | x | ------- dx | 2 | cos (x) | / 0
$$\int\limits_{0}^{\frac{\pi}{4}} \frac{x}{\cos^{2}{\left(x \right)}}\, dx$$
Integral(x/cos(x)^2, (x, 0, pi/4))
Respuesta (Indefinida)
[src]
/ / 2/x\\ / /x\\ / /x\\ 2/x\ / /x\\ 2/x\ / /x\\ 2/x\ / 2/x\\ /x\ | log|1 + tan |-|| log|1 + tan|-|| log|-1 + tan|-|| tan |-|*log|1 + tan|-|| tan |-|*log|-1 + tan|-|| tan |-|*log|1 + tan |-|| 2*x*tan|-| | x \ \2// \ \2// \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ | ------- dx = C + ---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - ------------ | 2 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ | cos (x) -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| | \2/ \2/ \2/ \2/ \2/ \2/ \2/ /
$$\int \frac{x}{\cos^{2}{\left(x \right)}}\, dx = C - \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$$
Gráfica
Respuesta
[src]
/ 2\ 2 2 2 / 2\
| / ___\ | / ___\ / ___\ / ___\ / / ___\\ / ___\ / ___\ / ___\ | / ___\ | / ___\
log\1 + \-1 + \/ 2 / / pi*I + log\2 - \/ 2 / log\\/ 2 / \-1 + \/ 2 / *\pi*I + log\2 - \/ 2 // \-1 + \/ 2 / *log\\/ 2 / \-1 + \/ 2 / *log\1 + \-1 + \/ 2 / / pi*\-1 + \/ 2 /
---------------------- - pi*I - --------------------- - ------------------ + ------------------------------------- + ------------------------ - ------------------------------------ - ----------------------
2 2 2 2 2 2 / 2\
/ ___\ / ___\ / ___\ / ___\ / ___\ / ___\ | / ___\ |
-1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / 2*\-1 + \-1 + \/ 2 / /
$$\frac{\log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} + \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\pi \left(-1 + \sqrt{2}\right)}{2 \left(-1 + \left(-1 + \sqrt{2}\right)^{2}\right)} - i \pi + \frac{\left(-1 + \sqrt{2}\right)^{2} \left(\log{\left(2 - \sqrt{2} \right)} + i \pi\right)}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(2 - \sqrt{2} \right)} + i \pi}{-1 + \left(-1 + \sqrt{2}\right)^{2}}$$
=
=
/ 2\ 2 2 2 / 2\
| / ___\ | / ___\ / ___\ / ___\ / / ___\\ / ___\ / ___\ / ___\ | / ___\ | / ___\
log\1 + \-1 + \/ 2 / / pi*I + log\2 - \/ 2 / log\\/ 2 / \-1 + \/ 2 / *\pi*I + log\2 - \/ 2 // \-1 + \/ 2 / *log\\/ 2 / \-1 + \/ 2 / *log\1 + \-1 + \/ 2 / / pi*\-1 + \/ 2 /
---------------------- - pi*I - --------------------- - ------------------ + ------------------------------------- + ------------------------ - ------------------------------------ - ----------------------
2 2 2 2 2 2 / 2\
/ ___\ / ___\ / ___\ / ___\ / ___\ / ___\ | / ___\ |
-1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / 2*\-1 + \-1 + \/ 2 / /
$$\frac{\log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} + \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\pi \left(-1 + \sqrt{2}\right)}{2 \left(-1 + \left(-1 + \sqrt{2}\right)^{2}\right)} - i \pi + \frac{\left(-1 + \sqrt{2}\right)^{2} \left(\log{\left(2 - \sqrt{2} \right)} + i \pi\right)}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(2 - \sqrt{2} \right)} + i \pi}{-1 + \left(-1 + \sqrt{2}\right)^{2}}$$
log(1 + (-1 + sqrt(2))^2)/(-1 + (-1 + sqrt(2))^2) - pi*i - (pi*i + log(2 - sqrt(2)))/(-1 + (-1 + sqrt(2))^2) - log(sqrt(2))/(-1 + (-1 + sqrt(2))^2) + (-1 + sqrt(2))^2*(pi*i + log(2 - sqrt(2)))/(-1 + (-1 + sqrt(2))^2) + (-1 + sqrt(2))^2*log(sqrt(2))/(-1 + (-1 + sqrt(2))^2) - (-1 + sqrt(2))^2*log(1 + (-1 + sqrt(2))^2)/(-1 + (-1 + sqrt(2))^2) - pi*(-1 + sqrt(2))/(2*(-1 + (-1 + sqrt(2))^2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.