Sr Examen
Integral de 1/(((x+1)^0,5)+x) dx
Solución
Ha introducido
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1 / | | 1 | ------------- dx | _______ | \/ x + 1 + x | / 0
$$\int\limits_{0}^{1} \frac{1}{x + \sqrt{x + 1}}\, dx$$
Integral(1/(sqrt(x + 1) + x), (x, 0, 1))
Respuesta (Indefinida)
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// / ___ /1 _______\\ \
|| |2*\/ 5 *|- + \/ x + 1 || |
|| ___ | \2 /| |
||-\/ 5 *acoth|-----------------------| 2 |
/ || \ 5 / /1 _______\ |
| ||-------------------------------------- for |- + \/ x + 1 | > 5/4|
| 1 || 10 \2 / | / _______\
| ------------- dx = C - 4*|< | + log\x + \/ x + 1 /
| _______ || / ___ /1 _______\\ |
| \/ x + 1 + x || |2*\/ 5 *|- + \/ x + 1 || |
| || ___ | \2 /| |
/ ||-\/ 5 *atanh|-----------------------| 2 |
|| \ 5 / /1 _______\ |
||-------------------------------------- for |- + \/ x + 1 | < 5/4|
\\ 10 \2 / /
$$\int \frac{1}{x + \sqrt{x + 1}}\, dx = C - 4 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(\sqrt{x + 1} + \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(\sqrt{x + 1} + \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(\sqrt{x + 1} + \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(\sqrt{x + 1} + \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) + \log{\left(x + \sqrt{x + 1} \right)}$$
Respuesta
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1 / | | / / / 2\ 2\ | | | | / ___\ | / ___\ | | | 2 | | \1 - \/ 5 / | \1 - \/ 5 / | | |------------------------------------ for Or|And|x >= -1, x < -1 + ------------|, x > -1 + ------------| | | / 2\ \ \ 4 / 4 / / ___\ | < | /1 _______\ | dx + log\1 + \/ 2 / | | | 4*|- + \/ 1 + x | | | | _______ | \2 / | | |5*\/ 1 + x *|1 - ------------------| | | \ 5 / | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{2}{5 \left(1 - \frac{4 \left(\sqrt{x + 1} + \frac{1}{2}\right)^{2}}{5}\right) \sqrt{x + 1}} & \text{for}\: \left(x \geq -1 \wedge x < -1 + \frac{\left(1 - \sqrt{5}\right)^{2}}{4}\right) \vee x > -1 + \frac{\left(1 - \sqrt{5}\right)^{2}}{4} \end{cases}\, dx + \log{\left(1 + \sqrt{2} \right)}$$
=
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1 / | | / / / 2\ 2\ | | | | / ___\ | / ___\ | | | 2 | | \1 - \/ 5 / | \1 - \/ 5 / | | |------------------------------------ for Or|And|x >= -1, x < -1 + ------------|, x > -1 + ------------| | | / 2\ \ \ 4 / 4 / / ___\ | < | /1 _______\ | dx + log\1 + \/ 2 / | | | 4*|- + \/ 1 + x | | | | _______ | \2 / | | |5*\/ 1 + x *|1 - ------------------| | | \ 5 / | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{2}{5 \left(1 - \frac{4 \left(\sqrt{x + 1} + \frac{1}{2}\right)^{2}}{5}\right) \sqrt{x + 1}} & \text{for}\: \left(x \geq -1 \wedge x < -1 + \frac{\left(1 - \sqrt{5}\right)^{2}}{4}\right) \vee x > -1 + \frac{\left(1 - \sqrt{5}\right)^{2}}{4} \end{cases}\, dx + \log{\left(1 + \sqrt{2} \right)}$$
Integral(Piecewise((2/(5*sqrt(1 + x)*(1 - 4*(1/2 + sqrt(1 + x))^2/5)), (x > -1 + (1 - sqrt(5))^2/4)∨((x >= -1)∧(x < -1 + (1 - sqrt(5))^2/4)))), (x, 0, 1)) + log(1 + sqrt(2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.