Sr Examen
Integral de xln|x^2-2x| dx
Solución
Ha introducido
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1 / | | /| 2 |\ | x*log\|x - 2*x|/ dx | / 0
$$\int\limits_{0}^{1} x \log{\left(\left|{x^{2} - 2 x}\right| \right)}\, dx$$
Integral(x*log(|x^2 - 2*x|), (x, 0, 1))
Respuesta (Indefinida)
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| | | | | | | | |
/ | 3 d / 2 \ | 3 d / 2 \ | 2 d / 2 \ | 2 d / 2 \ | d / 2 \ | d / 2 \ | d / 2 \ | 2 d / 2 \ | 2 d / 2 \
| | x*im (x)*--(im(x))*sign\x - 2*x/ | x*re (x)*--(re(x))*sign\x - 2*x/ | x*im (x)*--(re(x))*re(x)*sign\x - 2*x/ | x*re (x)*--(im(x))*im(x)*sign\x - 2*x/ | x*--(im(x))*im(x)*sign\x - 2*x/ | x*--(re(x))*re(x)*sign\x - 2*x/ | x*--(im(x))*im(x)*re(x)*sign\x - 2*x/ | x*re (x)*--(re(x))*sign\x - 2*x/ 2 /| 2 |\ | x*im (x)*--(re(x))*sign\x - 2*x/
| /| 2 |\ | dx | dx | dx | dx | dx | dx | dx | dx x *log\|x - 2*x|/ | dx
| x*log\|x - 2*x|/ dx = C - | --------------------------------- dx - | --------------------------------- dx - | --------------------------------------- dx - | --------------------------------------- dx - 2* | -------------------------------- dx - 2* | -------------------------------- dx + 2* | -------------------------------------- dx + 3* | --------------------------------- dx + ------------------ + | --------------------------------- dx
| | | 2 | | | 2 | | | 2 | | | 2 | | | 2 | | | 2 | | | 2 | | | 2 | 2 | | 2 |
/ | (-2 + x)*|x - 2*x| | (-2 + x)*|x - 2*x| | (-2 + x)*|x - 2*x| | (-2 + x)*|x - 2*x| | (-2 + x)*|x - 2*x| | (-2 + x)*|x - 2*x| | (-2 + x)*|x - 2*x| | (-2 + x)*|x - 2*x| | (-2 + x)*|x - 2*x|
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$$\int x \log{\left(\left|{x^{2} - 2 x}\right| \right)}\, dx = C + \frac{x^{2} \log{\left(\left|{x^{2} - 2 x}\right| \right)}}{2} - 2 \int \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx + 3 \int \frac{x \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - \int \frac{x \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - 2 \int \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx + \int \frac{x \left(\operatorname{im}{\left(x\right)}\right)^{2} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - \int \frac{x \left(\operatorname{im}{\left(x\right)}\right)^{3} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx + 2 \int \frac{x \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - \int \frac{x \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - \int \frac{x \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx$$
Respuesta
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1 / | | / 2 | | 2 / 2 \ x *(-2 + 2*x) 2 | | -1 - x - ------ + x*log\x - 2*x/ + ------------- for x - 2*x >= 0 | | -2 + x / 2 \ | | 2*\x - 2*x/ | < dx | | 2 | | 2 / 2 \ x *(2 - 2*x) | |-1 - x - ------ + x*log\- x + 2*x/ + -------------- otherwise | | -2 + x / 2 \ | \ 2*\- x + 2*x/ | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{x^{2} \left(2 x - 2\right)}{2 \left(x^{2} - 2 x\right)} + x \log{\left(x^{2} - 2 x \right)} - x - 1 - \frac{2}{x - 2} & \text{for}\: x^{2} - 2 x \geq 0 \\\frac{x^{2} \left(2 - 2 x\right)}{2 \left(- x^{2} + 2 x\right)} + x \log{\left(- x^{2} + 2 x \right)} - x - 1 - \frac{2}{x - 2} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / 2 | | 2 / 2 \ x *(-2 + 2*x) 2 | | -1 - x - ------ + x*log\x - 2*x/ + ------------- for x - 2*x >= 0 | | -2 + x / 2 \ | | 2*\x - 2*x/ | < dx | | 2 | | 2 / 2 \ x *(2 - 2*x) | |-1 - x - ------ + x*log\- x + 2*x/ + -------------- otherwise | | -2 + x / 2 \ | \ 2*\- x + 2*x/ | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{x^{2} \left(2 x - 2\right)}{2 \left(x^{2} - 2 x\right)} + x \log{\left(x^{2} - 2 x \right)} - x - 1 - \frac{2}{x - 2} & \text{for}\: x^{2} - 2 x \geq 0 \\\frac{x^{2} \left(2 - 2 x\right)}{2 \left(- x^{2} + 2 x\right)} + x \log{\left(- x^{2} + 2 x \right)} - x - 1 - \frac{2}{x - 2} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-1 - x - 2/(-2 + x) + x*log(x^2 - 2*x) + x^2*(-2 + 2*x)/(2*(x^2 - 2*x)), x^2 - 2*x >= 0), (-1 - x - 2/(-2 + x) + x*log(-x^2 + 2*x) + x^2*(2 - 2*x)/(2*(-x^2 + 2*x)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.