Sr Examen
Integral de 1/(x*(x^2-4*x+1)) dx
Solución
Ha introducido
[src]
2 / | | 1 | ---------------- dx | / 2 \ | x*\x - 4*x + 1/ | / 1
$$\int\limits_{1}^{2} \frac{1}{x \left(\left(x^{2} - 4 x\right) + 1\right)}\, dx$$
Integral(1/(x*(x^2 - 4*x + 1)), (x, 1, 2))
Respuesta (Indefinida)
[src]
// / ___ \ \
|| ___ |\/ 3 *(-2 + x)| |
||-\/ 3 *acoth|--------------| |
/ || \ 3 / 2 |
| ||----------------------------- for (-2 + x) > 3| / 2 \
| 1 || 3 | log\1 + x - 4*x/
| ---------------- dx = C + 2*|< | - ----------------- + log(x)
| / 2 \ || / ___ \ | 2
| x*\x - 4*x + 1/ || ___ |\/ 3 *(-2 + x)| |
| ||-\/ 3 *atanh|--------------| |
/ || \ 3 / 2 |
||----------------------------- for (-2 + x) < 3|
\\ 3 /
$$\int \frac{1}{x \left(\left(x^{2} - 4 x\right) + 1\right)}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} \left(x - 2\right)}{3} \right)}}{3} & \text{for}\: \left(x - 2\right)^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} \left(x - 2\right)}{3} \right)}}{3} & \text{for}\: \left(x - 2\right)^{2} < 3 \end{cases}\right) + \log{\left(x \right)} - \frac{\log{\left(x^{2} - 4 x + 1 \right)}}{2}$$
Gráfica
Respuesta
[src]
/ 2\ / / 2 \\ / 2\ / / 2 \\
| / ___\ | | | / ___\ || | / ___\ | | | / ___\ ||
| | 1 \/ 3 | | | | | 1 \/ 3 | || | | 1 \/ 3 | | | | | 1 \/ 3 | ||
/ ___\ | ___ 78*|- - - -----| | / ___\ | | 78*|- - + -----| ___|| / ___\ | ___ 78*|- - - -----| | / ___\ | | 78*|- - + -----| ___||
| 1 \/ 3 | | 91 3*\/ 3 \ 2 3 / | | 1 \/ 3 | | |91 \ 2 3 / 3*\/ 3 || | 1 \/ 3 | | 137 3*\/ 3 \ 2 3 / | | 1 \/ 3 | | |137 \ 2 3 / 3*\/ 3 ||
|- - - -----|*log|- -- - ------- + -----------------| + |- - + -----|*|pi*I + log|-- - ----------------- - -------|| - |- - - -----|*log|- --- - ------- + -----------------| - |- - + -----|*|pi*I + log|--- - ----------------- - -------|| + log(2)
\ 2 3 / \ 46 23 23 / \ 2 3 / \ \46 23 23 // \ 2 3 / \ 46 23 23 / \ 2 3 / \ \ 46 23 23 //
$$\left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right) \log{\left(- \frac{91}{46} - \frac{3 \sqrt{3}}{23} + \frac{78 \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right)^{2}}{23} \right)} - \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right) \log{\left(- \frac{137}{46} - \frac{3 \sqrt{3}}{23} + \frac{78 \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right)^{2}}{23} \right)} + \log{\left(2 \right)} - \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right) \left(\log{\left(- \frac{3 \sqrt{3}}{23} - \frac{78 \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right)^{2}}{23} + \frac{137}{46} \right)} + i \pi\right) + \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right) \left(\log{\left(- \frac{3 \sqrt{3}}{23} - \frac{78 \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right)^{2}}{23} + \frac{91}{46} \right)} + i \pi\right)$$
=
=
/ 2\ / / 2 \\ / 2\ / / 2 \\
| / ___\ | | | / ___\ || | / ___\ | | | / ___\ ||
| | 1 \/ 3 | | | | | 1 \/ 3 | || | | 1 \/ 3 | | | | | 1 \/ 3 | ||
/ ___\ | ___ 78*|- - - -----| | / ___\ | | 78*|- - + -----| ___|| / ___\ | ___ 78*|- - - -----| | / ___\ | | 78*|- - + -----| ___||
| 1 \/ 3 | | 91 3*\/ 3 \ 2 3 / | | 1 \/ 3 | | |91 \ 2 3 / 3*\/ 3 || | 1 \/ 3 | | 137 3*\/ 3 \ 2 3 / | | 1 \/ 3 | | |137 \ 2 3 / 3*\/ 3 ||
|- - - -----|*log|- -- - ------- + -----------------| + |- - + -----|*|pi*I + log|-- - ----------------- - -------|| - |- - - -----|*log|- --- - ------- + -----------------| - |- - + -----|*|pi*I + log|--- - ----------------- - -------|| + log(2)
\ 2 3 / \ 46 23 23 / \ 2 3 / \ \46 23 23 // \ 2 3 / \ 46 23 23 / \ 2 3 / \ \ 46 23 23 //
$$\left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right) \log{\left(- \frac{91}{46} - \frac{3 \sqrt{3}}{23} + \frac{78 \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right)^{2}}{23} \right)} - \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right) \log{\left(- \frac{137}{46} - \frac{3 \sqrt{3}}{23} + \frac{78 \left(- \frac{\sqrt{3}}{3} - \frac{1}{2}\right)^{2}}{23} \right)} + \log{\left(2 \right)} - \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right) \left(\log{\left(- \frac{3 \sqrt{3}}{23} - \frac{78 \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right)^{2}}{23} + \frac{137}{46} \right)} + i \pi\right) + \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right) \left(\log{\left(- \frac{3 \sqrt{3}}{23} - \frac{78 \left(- \frac{1}{2} + \frac{\sqrt{3}}{3}\right)^{2}}{23} + \frac{91}{46} \right)} + i \pi\right)$$
(-1/2 - sqrt(3)/3)*log(-91/46 - 3*sqrt(3)/23 + 78*(-1/2 - sqrt(3)/3)^2/23) + (-1/2 + sqrt(3)/3)*(pi*i + log(91/46 - 78*(-1/2 + sqrt(3)/3)^2/23 - 3*sqrt(3)/23)) - (-1/2 - sqrt(3)/3)*log(-137/46 - 3*sqrt(3)/23 + 78*(-1/2 - sqrt(3)/3)^2/23) - (-1/2 + sqrt(3)/3)*(pi*i + log(137/46 - 78*(-1/2 + sqrt(3)/3)^2/23 - 3*sqrt(3)/23)) + log(2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.