Sr Examen
Integral de (x+sqrt(2x+1))/(x-sqrt(2x+1))/ dx
Solución
Ha introducido
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1 / | | _________ | x + \/ 2*x + 1 | --------------- dx | _________ | x - \/ 2*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{x + \sqrt{2 x + 1}}{x - \sqrt{2 x + 1}}\, dx$$
Integral((x + sqrt(2*x + 1))/(x - sqrt(2*x + 1)), (x, 0, 1))
Respuesta (Indefinida)
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// / ___ / _________\\ \
|| ___ |\/ 2 *\-1 + \/ 2*x + 1 /| |
/ ||-\/ 2 *acoth|------------------------| 2 |
| || \ 2 / / _________\ |
| _________ ||--------------------------------------- for \-1 + \/ 2*x + 1 / > 2|
| x + \/ 2*x + 1 1 _________ / _________ \ || 2 |
| --------------- dx = - + C + x + 4*\/ 2*x + 1 + 4*log\- 2*\/ 2*x + 1 + 2*x/ + 12*|< |
| _________ 2 || / ___ / _________\\ |
| x - \/ 2*x + 1 || ___ |\/ 2 *\-1 + \/ 2*x + 1 /| |
| ||-\/ 2 *atanh|------------------------| 2 |
/ || \ 2 / / _________\ |
||--------------------------------------- for \-1 + \/ 2*x + 1 / < 2|
\\ 2 /
$$\int \frac{x + \sqrt{2 x + 1}}{x - \sqrt{2 x + 1}}\, dx = C + x + 4 \sqrt{2 x + 1} + 12 \left(\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} \left(\sqrt{2 x + 1} - 1\right)}{2} \right)}}{2} & \text{for}\: \left(\sqrt{2 x + 1} - 1\right)^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} \left(\sqrt{2 x + 1} - 1\right)}{2} \right)}}{2} & \text{for}\: \left(\sqrt{2 x + 1} - 1\right)^{2} < 2 \end{cases}\right) + 4 \log{\left(2 x - 2 \sqrt{2 x + 1} \right)} + \frac{1}{2}$$
Respuesta
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1
/
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| / / / 2\ 2\
| | | | / ___\ | / ___\ |
| | -6 | | 1 \1 + \/ 2 / | 1 \1 + \/ 2 / |
___ / ___\ | |------------------------------------- for Or|And|x >= -1/2, x < - - + ------------|, x > - - + ------------|
-3 - 4*log(2) + 4*\/ 3 + 4*log\-2 + 2*\/ 3 / + | < / 2\ \ \ 2 2 / 2 2 / dx
| | | / _________\ |
| | _________ | \-1 + \/ 1 + 2*x / |
| |\/ 1 + 2*x *|1 - -------------------|
| \ \ 2 /
|
/
0
$$\int\limits_{0}^{1} \begin{cases} - \frac{6}{\left(1 - \frac{\left(\sqrt{2 x + 1} - 1\right)^{2}}{2}\right) \sqrt{2 x + 1}} & \text{for}\: \left(x \geq - \frac{1}{2} \wedge x < - \frac{1}{2} + \frac{\left(1 + \sqrt{2}\right)^{2}}{2}\right) \vee x > - \frac{1}{2} + \frac{\left(1 + \sqrt{2}\right)^{2}}{2} \end{cases}\, dx - 3 - 4 \log{\left(2 \right)} + 4 \log{\left(-2 + 2 \sqrt{3} \right)} + 4 \sqrt{3}$$
=
=
1
/
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| / / / 2\ 2\
| | | | / ___\ | / ___\ |
| | -6 | | 1 \1 + \/ 2 / | 1 \1 + \/ 2 / |
___ / ___\ | |------------------------------------- for Or|And|x >= -1/2, x < - - + ------------|, x > - - + ------------|
-3 - 4*log(2) + 4*\/ 3 + 4*log\-2 + 2*\/ 3 / + | < / 2\ \ \ 2 2 / 2 2 / dx
| | | / _________\ |
| | _________ | \-1 + \/ 1 + 2*x / |
| |\/ 1 + 2*x *|1 - -------------------|
| \ \ 2 /
|
/
0
$$\int\limits_{0}^{1} \begin{cases} - \frac{6}{\left(1 - \frac{\left(\sqrt{2 x + 1} - 1\right)^{2}}{2}\right) \sqrt{2 x + 1}} & \text{for}\: \left(x \geq - \frac{1}{2} \wedge x < - \frac{1}{2} + \frac{\left(1 + \sqrt{2}\right)^{2}}{2}\right) \vee x > - \frac{1}{2} + \frac{\left(1 + \sqrt{2}\right)^{2}}{2} \end{cases}\, dx - 3 - 4 \log{\left(2 \right)} + 4 \log{\left(-2 + 2 \sqrt{3} \right)} + 4 \sqrt{3}$$
-3 - 4*log(2) + 4*sqrt(3) + 4*log(-2 + 2*sqrt(3)) + Integral(Piecewise((-6/(sqrt(1 + 2*x)*(1 - (-1 + sqrt(1 + 2*x))^2/2)), (x > -1/2 + (1 + sqrt(2))^2/2)∨((x >= -1/2)∧(x < -1/2 + (1 + sqrt(2))^2/2)))), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.