Sr Examen
Integral de x/((1+x^2)(sqrt(2+x^2))) dx
Solución
Ha introducido
[src]
1 / | | x | -------------------- dx | ________ | / 2\ / 2 | \1 + x /*\/ 2 + x | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} + 1\right) \sqrt{x^{2} + 2}}\, dx$$
Integral(x/(((1 + x^2)*sqrt(2 + x^2))), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ // / ________\ \ | || | / 2 | 2 | | x ||-acoth\\/ 2 + x / for 2 + x > 1| | -------------------- dx = C + |< | | ________ || / ________\ | | / 2\ / 2 || | / 2 | 2 | | \1 + x /*\/ 2 + x \\-atanh\\/ 2 + x / for 2 + x < 1/ | /
$$\int \frac{x}{\left(x^{2} + 1\right) \sqrt{x^{2} + 2}}\, dx = C + \begin{cases} - \operatorname{acoth}{\left(\sqrt{x^{2} + 2} \right)} & \text{for}\: x^{2} + 2 > 1 \\- \operatorname{atanh}{\left(\sqrt{x^{2} + 2} \right)} & \text{for}\: x^{2} + 2 < 1 \end{cases}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\ / ___\ / ___\
| \/ 2 | | \/ 3 | | \/ 2 | | \/ 3 |
log|1 + -----| log|1 - -----| log|1 - -----| log|1 + -----|
\ 2 / \ 3 / \ 2 / \ 3 /
-------------- + -------------- - -------------- - --------------
2 2 2 2
$$\frac{\log{\left(1 - \frac{\sqrt{3}}{3} \right)}}{2} - \frac{\log{\left(\frac{\sqrt{3}}{3} + 1 \right)}}{2} + \frac{\log{\left(\frac{\sqrt{2}}{2} + 1 \right)}}{2} - \frac{\log{\left(1 - \frac{\sqrt{2}}{2} \right)}}{2}$$
=
=
/ ___\ / ___\ / ___\ / ___\
| \/ 2 | | \/ 3 | | \/ 2 | | \/ 3 |
log|1 + -----| log|1 - -----| log|1 - -----| log|1 + -----|
\ 2 / \ 3 / \ 2 / \ 3 /
-------------- + -------------- - -------------- - --------------
2 2 2 2
$$\frac{\log{\left(1 - \frac{\sqrt{3}}{3} \right)}}{2} - \frac{\log{\left(\frac{\sqrt{3}}{3} + 1 \right)}}{2} + \frac{\log{\left(\frac{\sqrt{2}}{2} + 1 \right)}}{2} - \frac{\log{\left(1 - \frac{\sqrt{2}}{2} \right)}}{2}$$
log(1 + sqrt(2)/2)/2 + log(1 - sqrt(3)/3)/2 - log(1 - sqrt(2)/2)/2 - log(1 + sqrt(3)/3)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.