Sr Examen
Integral de (x^3)*dx/(√1+2x-x^2) dx
Solución
Ha introducido
[src]
1 / | | 3 | x | ---------------- dx | ___ 2 | \/ 1 + 2*x - x | / 0
$$\int\limits_{0}^{1} \frac{x^{3}}{- x^{2} + \left(2 x + \sqrt{1}\right)}\, dx$$
Integral(x^3/(sqrt(1) + 2*x - x^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ___ \ \
|| ___ |\/ 2 *(-1 + x)| |
/ ||-\/ 2 *acoth|--------------| |
| || \ 2 / 2 |
| 3 ||----------------------------- for (-1 + x) > 2| / 2 \ 2
| x || 2 | 5*log\-1 + x - 2*x/ x
| ---------------- dx = C - 7*|< | - 2*x - -------------------- - --
| ___ 2 || / ___ \ | 2 2
| \/ 1 + 2*x - x || ___ |\/ 2 *(-1 + x)| |
| ||-\/ 2 *atanh|--------------| |
/ || \ 2 / 2 |
||----------------------------- for (-1 + x) < 2|
\\ 2 /
$$\int \frac{x^{3}}{- x^{2} + \left(2 x + \sqrt{1}\right)}\, dx = C - \frac{x^{2}}{2} - 2 x - 7 \left(\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} \left(x - 1\right)}{2} \right)}}{2} & \text{for}\: \left(x - 1\right)^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} \left(x - 1\right)}{2} \right)}}{2} & \text{for}\: \left(x - 1\right)^{2} < 2 \end{cases}\right) - \frac{5 \log{\left(x^{2} - 2 x - 1 \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\ / ___\ / ___\ 5 |5 7*\/ 2 | / ___\ |5 7*\/ 2 | / / ___\\ |5 7*\/ 2 | / ___\ |5 7*\/ 2 | / / ___\\ - - + |- - -------|*log\-1 + \/ 2 / + |- + -------|*\pi*I + log\1 + \/ 2 // - |- - -------|*log\\/ 2 / - |- + -------|*\pi*I + log\\/ 2 // 2 \2 4 / \2 4 / \2 4 / \2 4 /
$$- \frac{5}{2} + \left(\frac{5}{2} - \frac{7 \sqrt{2}}{4}\right) \log{\left(-1 + \sqrt{2} \right)} - \left(\frac{5}{2} - \frac{7 \sqrt{2}}{4}\right) \log{\left(\sqrt{2} \right)} - \left(\frac{7 \sqrt{2}}{4} + \frac{5}{2}\right) \left(\log{\left(\sqrt{2} \right)} + i \pi\right) + \left(\frac{7 \sqrt{2}}{4} + \frac{5}{2}\right) \left(\log{\left(1 + \sqrt{2} \right)} + i \pi\right)$$
=
=
/ ___\ / ___\ / ___\ / ___\ 5 |5 7*\/ 2 | / ___\ |5 7*\/ 2 | / / ___\\ |5 7*\/ 2 | / ___\ |5 7*\/ 2 | / / ___\\ - - + |- - -------|*log\-1 + \/ 2 / + |- + -------|*\pi*I + log\1 + \/ 2 // - |- - -------|*log\\/ 2 / - |- + -------|*\pi*I + log\\/ 2 // 2 \2 4 / \2 4 / \2 4 / \2 4 /
$$- \frac{5}{2} + \left(\frac{5}{2} - \frac{7 \sqrt{2}}{4}\right) \log{\left(-1 + \sqrt{2} \right)} - \left(\frac{5}{2} - \frac{7 \sqrt{2}}{4}\right) \log{\left(\sqrt{2} \right)} - \left(\frac{7 \sqrt{2}}{4} + \frac{5}{2}\right) \left(\log{\left(\sqrt{2} \right)} + i \pi\right) + \left(\frac{7 \sqrt{2}}{4} + \frac{5}{2}\right) \left(\log{\left(1 + \sqrt{2} \right)} + i \pi\right)$$
-5/2 + (5/2 - 7*sqrt(2)/4)*log(-1 + sqrt(2)) + (5/2 + 7*sqrt(2)/4)*(pi*i + log(1 + sqrt(2))) - (5/2 - 7*sqrt(2)/4)*log(sqrt(2)) - (5/2 + 7*sqrt(2)/4)*(pi*i + log(sqrt(2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.