Sr Examen
Integral de (3x-1)/(4x^2-4x-7) dx
Solución
Ha introducido
[src]
1 / | | 3*x - 1 | -------------- dx | 2 | 4*x - 4*x - 7 | / 0
$$\int\limits_{0}^{1} \frac{3 x - 1}{\left(4 x^{2} - 4 x\right) - 7}\, dx$$
Integral((3*x - 1)/(4*x^2 - 4*x - 7), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ / ___ \
| ___ |\/ 2 *(-1/2 + x)|
|-\/ 2 *acoth|----------------|
| \ 2 / 2
|------------------------------- for (-1/2 + x) > 2
| 2
<
| / ___ \
| ___ |\/ 2 *(-1/2 + x)|
|-\/ 2 *atanh|----------------|
/ | \ 2 / 2
| |------------------------------- for (-1/2 + x) < 2 / 2\
| 3*x - 1 \ 2 3*log\-7 - 4*x + 4*x /
| -------------- dx = C + ----------------------------------------------------- + ----------------------
| 2 8 8
| 4*x - 4*x - 7
|
/
$$\int \frac{3 x - 1}{\left(4 x^{2} - 4 x\right) - 7}\, dx = C + \frac{\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} \left(x - \frac{1}{2}\right)}{2} \right)}}{2} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} \left(x - \frac{1}{2}\right)}{2} \right)}}{2} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < 2 \end{cases}}{8} + \frac{3 \log{\left(4 x^{2} - 4 x - 7 \right)}}{8}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\ / ___\ / ___\ |3 \/ 2 | /1 ___\ |3 \/ 2 | / / 1 ___\\ |3 \/ 2 | / 1 ___\ |3 \/ 2 | / /1 ___\\ |- - -----|*log|- + \/ 2 | + |- + -----|*|pi*I + log|- - + \/ 2 || - |- - -----|*log|- - + \/ 2 | - |- + -----|*|pi*I + log|- + \/ 2 || \8 32 / \2 / \8 32 / \ \ 2 // \8 32 / \ 2 / \8 32 / \ \2 //
$$- \left(\frac{3}{8} - \frac{\sqrt{2}}{32}\right) \log{\left(- \frac{1}{2} + \sqrt{2} \right)} + \left(\frac{3}{8} - \frac{\sqrt{2}}{32}\right) \log{\left(\frac{1}{2} + \sqrt{2} \right)} - \left(\frac{\sqrt{2}}{32} + \frac{3}{8}\right) \left(\log{\left(\frac{1}{2} + \sqrt{2} \right)} + i \pi\right) + \left(\frac{\sqrt{2}}{32} + \frac{3}{8}\right) \left(\log{\left(- \frac{1}{2} + \sqrt{2} \right)} + i \pi\right)$$
=
=
/ ___\ / ___\ / ___\ / ___\ |3 \/ 2 | /1 ___\ |3 \/ 2 | / / 1 ___\\ |3 \/ 2 | / 1 ___\ |3 \/ 2 | / /1 ___\\ |- - -----|*log|- + \/ 2 | + |- + -----|*|pi*I + log|- - + \/ 2 || - |- - -----|*log|- - + \/ 2 | - |- + -----|*|pi*I + log|- + \/ 2 || \8 32 / \2 / \8 32 / \ \ 2 // \8 32 / \ 2 / \8 32 / \ \2 //
$$- \left(\frac{3}{8} - \frac{\sqrt{2}}{32}\right) \log{\left(- \frac{1}{2} + \sqrt{2} \right)} + \left(\frac{3}{8} - \frac{\sqrt{2}}{32}\right) \log{\left(\frac{1}{2} + \sqrt{2} \right)} - \left(\frac{\sqrt{2}}{32} + \frac{3}{8}\right) \left(\log{\left(\frac{1}{2} + \sqrt{2} \right)} + i \pi\right) + \left(\frac{\sqrt{2}}{32} + \frac{3}{8}\right) \left(\log{\left(- \frac{1}{2} + \sqrt{2} \right)} + i \pi\right)$$
(3/8 - sqrt(2)/32)*log(1/2 + sqrt(2)) + (3/8 + sqrt(2)/32)*(pi*i + log(-1/2 + sqrt(2))) - (3/8 - sqrt(2)/32)*log(-1/2 + sqrt(2)) - (3/8 + sqrt(2)/32)*(pi*i + log(1/2 + sqrt(2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.