Sr Examen
Integral de dx/(√x^2+√2x+√5) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------------------ dx | 2 | ___ _____ ___ | \/ x + \/ 2*x + \/ 5 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(\left(\sqrt{x}\right)^{2} + \sqrt{2 x}\right) + \sqrt{5}}\, dx$$
Integral(1/((sqrt(x))^2 + sqrt(2*x) + sqrt(5)), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ ___ \
| ___ \/ 2 |
| \/ x + ----- |
___ | 2 |
\/ 2 *atan|-----------------|
| _____________|
/ | / 1 ___ |
| | / - - + \/ 5 |
| 1 \\/ 2 / / ___ ___ ___\
| ------------------------ dx = C - ----------------------------- + log\x + \/ 5 + \/ 2 *\/ x /
| 2 _____________
| ___ _____ ___ / 1 ___
| \/ x + \/ 2*x + \/ 5 / - - + \/ 5
| \/ 2
/
$$\int \frac{1}{\left(\left(\sqrt{x}\right)^{2} + \sqrt{2 x}\right) + \sqrt{5}}\, dx = C + \log{\left(\sqrt{2} \sqrt{x} + x + \sqrt{5} \right)} - \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{x} + \frac{\sqrt{2}}{2}}{\sqrt{- \frac{1}{2} + \sqrt{5}}} \right)}}{\sqrt{- \frac{1}{2} + \sqrt{5}}}$$
Gráfica
Respuesta
[src]
/ _____________\ / _____________\ / _____________\ / _____________\
| ___ ___ / ___ | | ___ ___ / ___ | | ___ ___ / ___ | | ___ ___ / ___ |
/ _____________\ / _____________\ |\/ 2 \/ 2 *\/ 1 - 2*\/ 5 | | \/ 2 \/ 2 *\/ 1 - 2*\/ 5 | |\/ 2 \/ 2 *\/ 1 - 2*\/ 5 | | \/ 2 \/ 2 *\/ 1 - 2*\/ 5 | / _____________\ / _____________\
| ___ ___ / ___ | | ___ ___ / ___ | log|----- - ----------------------| log|1 + ----- + ----------------------| log|----- + ----------------------| log|1 + ----- - ----------------------| | ___ ___ / ___ | | ___ ___ / ___ |
|\/ 2 \/ 2 *\/ 1 - 2*\/ 5 | |\/ 2 \/ 2 *\/ 1 - 2*\/ 5 | \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 / | \/ 2 \/ 2 *\/ 1 - 2*\/ 5 | | \/ 2 \/ 2 *\/ 1 - 2*\/ 5 |
- log|----- + ----------------------| - log|----- - ----------------------| + ----------------------------------- + --------------------------------------- - ----------------------------------- - --------------------------------------- + log|1 + ----- + ----------------------| + log|1 + ----- - ----------------------|
\ 2 2 / \ 2 2 / _____________ _____________ _____________ _____________ \ 2 2 / \ 2 2 /
/ ___ / ___ / ___ / ___
\/ 1 - 2*\/ 5 \/ 1 - 2*\/ 5 \/ 1 - 2*\/ 5 \/ 1 - 2*\/ 5
$$- \log{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)} + \log{\left(\frac{\sqrt{2}}{2} + 1 - \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)} + \frac{\log{\left(\frac{\sqrt{2}}{2} + 1 + \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}}{\sqrt{1 - 2 \sqrt{5}}} + \frac{\log{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}}{\sqrt{1 - 2 \sqrt{5}}} - \frac{\log{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}}{\sqrt{1 - 2 \sqrt{5}}} - \frac{\log{\left(\frac{\sqrt{2}}{2} + 1 - \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}}{\sqrt{1 - 2 \sqrt{5}}} + \log{\left(\frac{\sqrt{2}}{2} + 1 + \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)} - \log{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}$$
=
=
/ _____________\ / _____________\ / _____________\ / _____________\
| ___ ___ / ___ | | ___ ___ / ___ | | ___ ___ / ___ | | ___ ___ / ___ |
/ _____________\ / _____________\ |\/ 2 \/ 2 *\/ 1 - 2*\/ 5 | | \/ 2 \/ 2 *\/ 1 - 2*\/ 5 | |\/ 2 \/ 2 *\/ 1 - 2*\/ 5 | | \/ 2 \/ 2 *\/ 1 - 2*\/ 5 | / _____________\ / _____________\
| ___ ___ / ___ | | ___ ___ / ___ | log|----- - ----------------------| log|1 + ----- + ----------------------| log|----- + ----------------------| log|1 + ----- - ----------------------| | ___ ___ / ___ | | ___ ___ / ___ |
|\/ 2 \/ 2 *\/ 1 - 2*\/ 5 | |\/ 2 \/ 2 *\/ 1 - 2*\/ 5 | \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 / | \/ 2 \/ 2 *\/ 1 - 2*\/ 5 | | \/ 2 \/ 2 *\/ 1 - 2*\/ 5 |
- log|----- + ----------------------| - log|----- - ----------------------| + ----------------------------------- + --------------------------------------- - ----------------------------------- - --------------------------------------- + log|1 + ----- + ----------------------| + log|1 + ----- - ----------------------|
\ 2 2 / \ 2 2 / _____________ _____________ _____________ _____________ \ 2 2 / \ 2 2 /
/ ___ / ___ / ___ / ___
\/ 1 - 2*\/ 5 \/ 1 - 2*\/ 5 \/ 1 - 2*\/ 5 \/ 1 - 2*\/ 5
$$- \log{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)} + \log{\left(\frac{\sqrt{2}}{2} + 1 - \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)} + \frac{\log{\left(\frac{\sqrt{2}}{2} + 1 + \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}}{\sqrt{1 - 2 \sqrt{5}}} + \frac{\log{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}}{\sqrt{1 - 2 \sqrt{5}}} - \frac{\log{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}}{\sqrt{1 - 2 \sqrt{5}}} - \frac{\log{\left(\frac{\sqrt{2}}{2} + 1 - \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}}{\sqrt{1 - 2 \sqrt{5}}} + \log{\left(\frac{\sqrt{2}}{2} + 1 + \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)} - \log{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} \sqrt{1 - 2 \sqrt{5}}}{2} \right)}$$
-log(sqrt(2)/2 + sqrt(2)*sqrt(1 - 2*sqrt(5))/2) - log(sqrt(2)/2 - sqrt(2)*sqrt(1 - 2*sqrt(5))/2) + log(sqrt(2)/2 - sqrt(2)*sqrt(1 - 2*sqrt(5))/2)/sqrt(1 - 2*sqrt(5)) + log(1 + sqrt(2)/2 + sqrt(2)*sqrt(1 - 2*sqrt(5))/2)/sqrt(1 - 2*sqrt(5)) - log(sqrt(2)/2 + sqrt(2)*sqrt(1 - 2*sqrt(5))/2)/sqrt(1 - 2*sqrt(5)) - log(1 + sqrt(2)/2 - sqrt(2)*sqrt(1 - 2*sqrt(5))/2)/sqrt(1 - 2*sqrt(5)) + log(1 + sqrt(2)/2 + sqrt(2)*sqrt(1 - 2*sqrt(5))/2) + log(1 + sqrt(2)/2 - sqrt(2)*sqrt(1 - 2*sqrt(5))/2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.