Sr Examen
Integral de sqrt(x+2)/(x-2(sqrt(x+2))) dx
Solución
Ha introducido
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1 / | | _______ | \/ x + 2 | --------------- dx | _______ | x - 2*\/ x + 2 | / 0
$$\int\limits_{0}^{1} \frac{\sqrt{x + 2}}{x - 2 \sqrt{x + 2}}\, dx$$
Integral(sqrt(x + 2)/(x - 2*sqrt(x + 2)), (x, 0, 1))
Respuesta (Indefinida)
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// / ___ / _______\\ \
|| ___ |\/ 3 *\-1 + \/ 2 + x /| |
/ ||-\/ 3 *acoth|----------------------| 2 |
| || \ 3 / / _______\ |
| _______ ||------------------------------------- for \-1 + \/ 2 + x / > 3|
| \/ x + 2 _______ / _______\ || 3 |
| --------------- dx = C + 2*\/ 2 + x + 2*log\x - 2*\/ 2 + x / + 8*|< |
| _______ || / ___ / _______\\ |
| x - 2*\/ x + 2 || ___ |\/ 3 *\-1 + \/ 2 + x /| |
| ||-\/ 3 *atanh|----------------------| 2 |
/ || \ 3 / / _______\ |
||------------------------------------- for \-1 + \/ 2 + x / < 3|
\\ 3 /
$$\int \frac{\sqrt{x + 2}}{x - 2 \sqrt{x + 2}}\, dx = C + 2 \sqrt{x + 2} + 8 \left(\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} \left(\sqrt{x + 2} - 1\right)}{3} \right)}}{3} & \text{for}\: \left(\sqrt{x + 2} - 1\right)^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} \left(\sqrt{x + 2} - 1\right)}{3} \right)}}{3} & \text{for}\: \left(\sqrt{x + 2} - 1\right)^{2} < 3 \end{cases}\right) + 2 \log{\left(x - 2 \sqrt{x + 2} \right)}$$
Respuesta
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1
/
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| / / / 2\ 2\
| | -4 | | / ___\ | / ___\ |
| |----------------------------------- for Or\And\x >= -2, x < -2 + \1 + \/ 3 / /, x > -2 + \1 + \/ 3 / /
___ / ___\ ___ / ___\ | | / 2\
- 2*\/ 2 - 2*log\2*\/ 2 / + 2*\/ 3 + 2*log\-1 + 2*\/ 3 / + | < | / _______\ | dx
| | | \-1 + \/ 2 + x / | _______
| |3*|1 - -----------------|*\/ 2 + x
| | \ 3 /
| \
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/
0
$$- 2 \sqrt{2} - 2 \log{\left(2 \sqrt{2} \right)} + \int\limits_{0}^{1} \begin{cases} - \frac{4}{3 \left(1 - \frac{\left(\sqrt{x + 2} - 1\right)^{2}}{3}\right) \sqrt{x + 2}} & \text{for}\: \left(x \geq -2 \wedge x < -2 + \left(1 + \sqrt{3}\right)^{2}\right) \vee x > -2 + \left(1 + \sqrt{3}\right)^{2} \end{cases}\, dx + 2 \log{\left(-1 + 2 \sqrt{3} \right)} + 2 \sqrt{3}$$
=
=
1
/
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| / / / 2\ 2\
| | -4 | | / ___\ | / ___\ |
| |----------------------------------- for Or\And\x >= -2, x < -2 + \1 + \/ 3 / /, x > -2 + \1 + \/ 3 / /
___ / ___\ ___ / ___\ | | / 2\
- 2*\/ 2 - 2*log\2*\/ 2 / + 2*\/ 3 + 2*log\-1 + 2*\/ 3 / + | < | / _______\ | dx
| | | \-1 + \/ 2 + x / | _______
| |3*|1 - -----------------|*\/ 2 + x
| | \ 3 /
| \
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/
0
$$- 2 \sqrt{2} - 2 \log{\left(2 \sqrt{2} \right)} + \int\limits_{0}^{1} \begin{cases} - \frac{4}{3 \left(1 - \frac{\left(\sqrt{x + 2} - 1\right)^{2}}{3}\right) \sqrt{x + 2}} & \text{for}\: \left(x \geq -2 \wedge x < -2 + \left(1 + \sqrt{3}\right)^{2}\right) \vee x > -2 + \left(1 + \sqrt{3}\right)^{2} \end{cases}\, dx + 2 \log{\left(-1 + 2 \sqrt{3} \right)} + 2 \sqrt{3}$$
-2*sqrt(2) - 2*log(2*sqrt(2)) + 2*sqrt(3) + 2*log(-1 + 2*sqrt(3)) + Integral(Piecewise((-4/(3*(1 - (-1 + sqrt(2 + x))^2/3)*sqrt(2 + x)), (x > -2 + (1 + sqrt(3))^2)∨((x >= -2)∧(x < -2 + (1 + sqrt(3))^2)))), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.