Sr Examen
Integral de (x+1)/(2^3-3x^2+1) dx
Solución
Ha introducido
[src]
1 / | | x + 1 | ------------ dx | 2 | 8 - 3*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{x + 1}{\left(8 - 3 x^{2}\right) + 1}\, dx$$
Integral((x + 1)/(8 - 3*x^2 + 1), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ / ___\
| ___ |x*\/ 3 |
|-\/ 3 *acoth|-------|
| \ 3 / 2
|---------------------- for x > 3
| 3
<
| / ___\
| ___ |x*\/ 3 |
|-\/ 3 *atanh|-------|
/ | \ 3 / 2
| |---------------------- for x < 3 / 2\
| x + 1 \ 3 log\-3 + x /
| ------------ dx = C - ----------------------------------- - ------------
| 2 3 6
| 8 - 3*x + 1
|
/
$$\int \frac{x + 1}{\left(8 - 3 x^{2}\right) + 1}\, dx = C - \frac{\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} < 3 \end{cases}}{3} - \frac{\log{\left(x^{2} - 3 \right)}}{6}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\ / ___\ / ___\ |1 \/ 3 | / ___\ |1 \/ 3 | / / ___\\ |1 \/ 3 | / ___\ |1 \/ 3 | / / ___\\ |- - -----|*log\\/ 3 / + |- + -----|*\pi*I + log\\/ 3 // - |- - -----|*log\1 + \/ 3 / - |- + -----|*\pi*I + log\-1 + \/ 3 // \6 18 / \6 18 / \6 18 / \6 18 /
$$- \left(\frac{1}{6} - \frac{\sqrt{3}}{18}\right) \log{\left(1 + \sqrt{3} \right)} + \left(\frac{1}{6} - \frac{\sqrt{3}}{18}\right) \log{\left(\sqrt{3} \right)} - \left(\frac{\sqrt{3}}{18} + \frac{1}{6}\right) \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right) + \left(\frac{\sqrt{3}}{18} + \frac{1}{6}\right) \left(\log{\left(\sqrt{3} \right)} + i \pi\right)$$
=
=
/ ___\ / ___\ / ___\ / ___\ |1 \/ 3 | / ___\ |1 \/ 3 | / / ___\\ |1 \/ 3 | / ___\ |1 \/ 3 | / / ___\\ |- - -----|*log\\/ 3 / + |- + -----|*\pi*I + log\\/ 3 // - |- - -----|*log\1 + \/ 3 / - |- + -----|*\pi*I + log\-1 + \/ 3 // \6 18 / \6 18 / \6 18 / \6 18 /
$$- \left(\frac{1}{6} - \frac{\sqrt{3}}{18}\right) \log{\left(1 + \sqrt{3} \right)} + \left(\frac{1}{6} - \frac{\sqrt{3}}{18}\right) \log{\left(\sqrt{3} \right)} - \left(\frac{\sqrt{3}}{18} + \frac{1}{6}\right) \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right) + \left(\frac{\sqrt{3}}{18} + \frac{1}{6}\right) \left(\log{\left(\sqrt{3} \right)} + i \pi\right)$$
(1/6 - sqrt(3)/18)*log(sqrt(3)) + (1/6 + sqrt(3)/18)*(pi*i + log(sqrt(3))) - (1/6 - sqrt(3)/18)*log(1 + sqrt(3)) - (1/6 + sqrt(3)/18)*(pi*i + log(-1 + sqrt(3)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.