Sr Examen
Integral de (1-2*x)/(x^(4)-2x^(2)-3) dx
Solución
Ha introducido
[src]
1 / | | 1 - 2*x | ------------- dx | 4 2 | x - 2*x - 3 | / 0
$$\int\limits_{0}^{1} \frac{1 - 2 x}{\left(x^{4} - 2 x^{2}\right) - 3}\, dx$$
Integral((1 - 2*x)/(x^4 - 2*x^2 - 3), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ / ___\
| ___ |x*\/ 3 |
|-\/ 3 *acoth|-------|
| \ 3 / 2
|---------------------- for x > 3
| 3
<
| / ___\
| ___ |x*\/ 3 |
|-\/ 3 *atanh|-------|
/ | \ 3 / 2
| / 2\ |---------------------- for x < 3 / 2\
| 1 - 2*x atan(x) log\-3 + x / \ 3 log\1 + x /
| ------------- dx = C - ------- - ------------ + ----------------------------------- + -----------
| 4 2 4 4 4 4
| x - 2*x - 3
|
/
$$\int \frac{1 - 2 x}{\left(x^{4} - 2 x^{2}\right) - 3}\, dx = C + \frac{\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} < 3 \end{cases}}{4} - \frac{\log{\left(x^{2} - 3 \right)}}{4} + \frac{\log{\left(x^{2} + 1 \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{4}$$
Gráfica
Respuesta
[src]
/ / 3 2 \\ / 2 3\ / / 3 2 \\ / 2 3\
| | / ___\ / ___\ || | / ___\ / ___\ | | | / ___\ / ___\ || | / ___\ / ___\ |
| | |1 \/ 3 | |1 \/ 3 | || | |1 \/ 3 | |1 \/ 3 | | | | |1 \/ 3 | |1 \/ 3 | || | |1 \/ 3 | |1 \/ 3 | |
/ ___\ | | 19200*|- - -----| 2112*|- - -----| ___|| / ___\ | ___ 2112*|- + -----| 19200*|- + -----| | / ___\ | | 19200*|- - -----| 2112*|- - -----| ___|| / ___\ | ___ 2112*|- + -----| 19200*|- + -----| |
pi log(2) |1 \/ 3 | | |518 \4 24 / \4 24 / 247*\/ 3 || |1 \/ 3 | | 518 247*\/ 3 \4 24 / \4 24 / | |1 \/ 3 | | |361 \4 24 / \4 24 / 247*\/ 3 || |1 \/ 3 | | 361 247*\/ 3 \4 24 / \4 24 / |
- -- + ------ + |- - -----|*|pi*I + log|--- - ------------------ - ----------------- - ---------|| + |- + -----|*log|- --- - --------- + ----------------- + ------------------| - |- - -----|*|pi*I + log|--- - ------------------ - ----------------- - ---------|| - |- + -----|*log|- --- - --------- + ----------------- + ------------------|
16 4 \4 24 / \ \157 157 157 942 // \4 24 / \ 157 942 157 157 / \4 24 / \ \157 157 157 942 // \4 24 / \ 157 942 157 157 /
$$- \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right) \log{\left(- \frac{361}{157} - \frac{247 \sqrt{3}}{942} + \frac{2112 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{2}}{157} + \frac{19200 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{3}}{157} \right)} - \frac{\pi}{16} + \frac{\log{\left(2 \right)}}{4} + \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right) \log{\left(- \frac{518}{157} - \frac{247 \sqrt{3}}{942} + \frac{2112 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{2}}{157} + \frac{19200 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{3}}{157} \right)} - \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right) \left(\log{\left(- \frac{19200 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{3}}{157} - \frac{247 \sqrt{3}}{942} - \frac{2112 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{2}}{157} + \frac{361}{157} \right)} + i \pi\right) + \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right) \left(\log{\left(- \frac{19200 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{3}}{157} - \frac{247 \sqrt{3}}{942} - \frac{2112 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{2}}{157} + \frac{518}{157} \right)} + i \pi\right)$$
=
=
/ / 3 2 \\ / 2 3\ / / 3 2 \\ / 2 3\
| | / ___\ / ___\ || | / ___\ / ___\ | | | / ___\ / ___\ || | / ___\ / ___\ |
| | |1 \/ 3 | |1 \/ 3 | || | |1 \/ 3 | |1 \/ 3 | | | | |1 \/ 3 | |1 \/ 3 | || | |1 \/ 3 | |1 \/ 3 | |
/ ___\ | | 19200*|- - -----| 2112*|- - -----| ___|| / ___\ | ___ 2112*|- + -----| 19200*|- + -----| | / ___\ | | 19200*|- - -----| 2112*|- - -----| ___|| / ___\ | ___ 2112*|- + -----| 19200*|- + -----| |
pi log(2) |1 \/ 3 | | |518 \4 24 / \4 24 / 247*\/ 3 || |1 \/ 3 | | 518 247*\/ 3 \4 24 / \4 24 / | |1 \/ 3 | | |361 \4 24 / \4 24 / 247*\/ 3 || |1 \/ 3 | | 361 247*\/ 3 \4 24 / \4 24 / |
- -- + ------ + |- - -----|*|pi*I + log|--- - ------------------ - ----------------- - ---------|| + |- + -----|*log|- --- - --------- + ----------------- + ------------------| - |- - -----|*|pi*I + log|--- - ------------------ - ----------------- - ---------|| - |- + -----|*log|- --- - --------- + ----------------- + ------------------|
16 4 \4 24 / \ \157 157 157 942 // \4 24 / \ 157 942 157 157 / \4 24 / \ \157 157 157 942 // \4 24 / \ 157 942 157 157 /
$$- \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right) \log{\left(- \frac{361}{157} - \frac{247 \sqrt{3}}{942} + \frac{2112 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{2}}{157} + \frac{19200 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{3}}{157} \right)} - \frac{\pi}{16} + \frac{\log{\left(2 \right)}}{4} + \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right) \log{\left(- \frac{518}{157} - \frac{247 \sqrt{3}}{942} + \frac{2112 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{2}}{157} + \frac{19200 \left(\frac{\sqrt{3}}{24} + \frac{1}{4}\right)^{3}}{157} \right)} - \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right) \left(\log{\left(- \frac{19200 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{3}}{157} - \frac{247 \sqrt{3}}{942} - \frac{2112 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{2}}{157} + \frac{361}{157} \right)} + i \pi\right) + \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right) \left(\log{\left(- \frac{19200 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{3}}{157} - \frac{247 \sqrt{3}}{942} - \frac{2112 \left(\frac{1}{4} - \frac{\sqrt{3}}{24}\right)^{2}}{157} + \frac{518}{157} \right)} + i \pi\right)$$
-pi/16 + log(2)/4 + (1/4 - sqrt(3)/24)*(pi*i + log(518/157 - 19200*(1/4 - sqrt(3)/24)^3/157 - 2112*(1/4 - sqrt(3)/24)^2/157 - 247*sqrt(3)/942)) + (1/4 + sqrt(3)/24)*log(-518/157 - 247*sqrt(3)/942 + 2112*(1/4 + sqrt(3)/24)^2/157 + 19200*(1/4 + sqrt(3)/24)^3/157) - (1/4 - sqrt(3)/24)*(pi*i + log(361/157 - 19200*(1/4 - sqrt(3)/24)^3/157 - 2112*(1/4 - sqrt(3)/24)^2/157 - 247*sqrt(3)/942)) - (1/4 + sqrt(3)/24)*log(-361/157 - 247*sqrt(3)/942 + 2112*(1/4 + sqrt(3)/24)^2/157 + 19200*(1/4 + sqrt(3)/24)^3/157)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.