Sr Examen
Integral de xdx/sin(x)^2 dx
Solución
Ha introducido
[src]
pi -- 3 / | | x | ------- dx | 2 | sin (x) | / pi -- 4
$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{x}{\sin^{2}{\left(x \right)}}\, dx$$
Integral(x/sin(x)^2, (x, pi/4, pi/3))
Respuesta (Indefinida)
[src]
/ /x\ | x*tan|-| | x / 2/x\\ \2/ x / /x\\ | ------- dx = C - log|1 + tan |-|| + -------- - -------- + log|tan|-|| | 2 \ \2// 2 /x\ \ \2// | sin (x) 2*tan|-| | \2/ /
$$\int \frac{x}{\sin^{2}{\left(x \right)}}\, dx = C + \frac{x \tan{\left(\frac{x}{2} \right)}}{2} - \frac{x}{2 \tan{\left(\frac{x}{2} \right)}} - \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}$$
Gráfica
Respuesta
[src]
/ ___\ ___ / ___\ / 2\
/ ___\ pi*\-1 + \/ 2 / pi*\/ 3 pi |\/ 3 | | / ___\ |
-log(4/3) - log\-1 + \/ 2 / - --------------- - -------- + -------------- + log|-----| + log\1 + \-1 + \/ 2 / /
8 9 / ___\ \ 3 /
8*\-1 + \/ 2 /
$$- \frac{\sqrt{3} \pi}{9} + \log{\left(\frac{\sqrt{3}}{3} \right)} - \log{\left(\frac{4}{3} \right)} - \frac{\pi \left(-1 + \sqrt{2}\right)}{8} + \log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)} - \log{\left(-1 + \sqrt{2} \right)} + \frac{\pi}{8 \left(-1 + \sqrt{2}\right)}$$
=
=
/ ___\ ___ / ___\ / 2\
/ ___\ pi*\-1 + \/ 2 / pi*\/ 3 pi |\/ 3 | | / ___\ |
-log(4/3) - log\-1 + \/ 2 / - --------------- - -------- + -------------- + log|-----| + log\1 + \-1 + \/ 2 / /
8 9 / ___\ \ 3 /
8*\-1 + \/ 2 /
$$- \frac{\sqrt{3} \pi}{9} + \log{\left(\frac{\sqrt{3}}{3} \right)} - \log{\left(\frac{4}{3} \right)} - \frac{\pi \left(-1 + \sqrt{2}\right)}{8} + \log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)} - \log{\left(-1 + \sqrt{2} \right)} + \frac{\pi}{8 \left(-1 + \sqrt{2}\right)}$$
-log(4/3) - log(-1 + sqrt(2)) - pi*(-1 + sqrt(2))/8 - pi*sqrt(3)/9 + pi/(8*(-1 + sqrt(2))) + log(sqrt(3)/3) + log(1 + (-1 + sqrt(2))^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.