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