Sr Examen
Integral de (√1+(√x))/(√(x³+1)) dx
Solución
Ha introducido
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1 / | | ___ ___ | \/ 1 + \/ x | ------------- dx | ________ | / 3 | \/ x + 1 | / 0
$$\int\limits_{0}^{1} \frac{\sqrt{x} + \sqrt{1}}{\sqrt{x^{3} + 1}}\, dx$$
Integral((sqrt(1) + sqrt(x))/sqrt(x^3 + 1), (x, 0, 1))
Respuesta (Indefinida)
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/ _ | |_ /1/3, 1/2 | 3 pi*I\ | ___ ___ / 3/2\ x*Gamma(1/3)* | | | x *e | | \/ 1 + \/ x 2*asinh\x / 2 1 \ 4/3 | / | ------------- dx = C + ------------- + --------------------------------------- | ________ 3 3*Gamma(4/3) | / 3 | \/ x + 1 | /
$$\int \frac{\sqrt{x} + \sqrt{1}}{\sqrt{x^{3} + 1}}\, dx = C + \frac{x \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {x^{3} e^{i \pi}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{2 \operatorname{asinh}{\left(x^{\frac{3}{2}} \right)}}{3}$$
Gráfica
Respuesta
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_
|_ /1/3, 1/2 | \
/ ___\ Gamma(1/3)* | | | -1|
2*log\1 + \/ 2 / 2 1 \ 4/3 | /
---------------- + -------------------------------
3 3*Gamma(4/3)
$$\frac{\Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {-1} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{2 \log{\left(1 + \sqrt{2} \right)}}{3}$$
=
=
_
|_ /1/3, 1/2 | \
/ ___\ Gamma(1/3)* | | | -1|
2*log\1 + \/ 2 / 2 1 \ 4/3 | /
---------------- + -------------------------------
3 3*Gamma(4/3)
$$\frac{\Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {-1} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{2 \log{\left(1 + \sqrt{2} \right)}}{3}$$
2*log(1 + sqrt(2))/3 + gamma(1/3)*hyper((1/3, 1/2), (4/3,), -1)/(3*gamma(4/3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.