Sr Examen
Integral de 1/(1+x²)^n dx
Solución
Ha introducido
[src]
oo / | | 1 | --------- dx | n | / 2\ | \1 + x / | / 0
$$\int\limits_{0}^{\infty} \frac{1}{\left(x^{2} + 1\right)^{n}}\, dx$$
Integral(1/((1 + x^2)^n), (x, 0, oo))
Respuesta (Indefinida)
[src]
/ | _ | 1 |_ /1/2, n | 2 pi*I\ | --------- dx = C + x* | | | x *e | | n 2 1 \ 3/2 | / | / 2\ | \1 + x / | /
$$\int \frac{1}{\left(x^{2} + 1\right)^{n}}\, dx = C + x {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, n \\ \frac{3}{2} \end{matrix}\middle| {x^{2} e^{i \pi}} \right)}$$
Respuesta
[src]
/ ____ |\/ pi *Gamma(-1/2 + n) |---------------------- for 1/2 + re(n) > 1 | 2*Gamma(n) | | oo | / < | | | -n | | / 2\ | | \1 + x / dx otherwise | | | / | 0 \
$$\begin{cases} \frac{\sqrt{\pi} \Gamma\left(n - \frac{1}{2}\right)}{2 \Gamma\left(n\right)} & \text{for}\: \operatorname{re}{\left(n\right)} + \frac{1}{2} > 1 \\\int\limits_{0}^{\infty} \left(x^{2} + 1\right)^{- n}\, dx & \text{otherwise} \end{cases}$$
=
=
/ ____ |\/ pi *Gamma(-1/2 + n) |---------------------- for 1/2 + re(n) > 1 | 2*Gamma(n) | | oo | / < | | | -n | | / 2\ | | \1 + x / dx otherwise | | | / | 0 \
$$\begin{cases} \frac{\sqrt{\pi} \Gamma\left(n - \frac{1}{2}\right)}{2 \Gamma\left(n\right)} & \text{for}\: \operatorname{re}{\left(n\right)} + \frac{1}{2} > 1 \\\int\limits_{0}^{\infty} \left(x^{2} + 1\right)^{- n}\, dx & \text{otherwise} \end{cases}$$
Piecewise((sqrt(pi)*gamma(-1/2 + n)/(2*gamma(n)), 1/2 + re(n) > 1), (Integral((1 + x^2)^(-n), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.