Sr Examen
Integral de 1/((√5+2x-x^2)) dx
Solución
Ha introducido
[src]
1 / | | 1 | ---------------- dx | ___ 2 | \/ 5 + 2*x - x | / 0
$$\int\limits_{0}^{1} \frac{1}{- x^{2} + \left(2 x + \sqrt{5}\right)}\, dx$$
Integral(1/(sqrt(5) + 2*x - x^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / -1 + x \ \
||-acoth|--------------| |
|| | ___________| |
|| | / ___ | |
|| \\/ 1 + \/ 5 / 2 ___|
||----------------------- for (-1 + x) > 1 + \/ 5 |
/ || ___________ |
| || / ___ |
| 1 || \/ 1 + \/ 5 |
| ---------------- dx = C - |< |
| ___ 2 || / -1 + x \ |
| \/ 5 + 2*x - x ||-atanh|--------------| |
| || | ___________| |
/ || | / ___ | |
|| \\/ 1 + \/ 5 / 2 ___|
||----------------------- for (-1 + x) < 1 + \/ 5 |
|| ___________ |
|| / ___ |
\\ \/ 1 + \/ 5 /
$$\int \frac{1}{- x^{2} + \left(2 x + \sqrt{5}\right)}\, dx = C - \begin{cases} - \frac{\operatorname{acoth}{\left(\frac{x - 1}{\sqrt{1 + \sqrt{5}}} \right)}}{\sqrt{1 + \sqrt{5}}} & \text{for}\: \left(x - 1\right)^{2} > 1 + \sqrt{5} \\- \frac{\operatorname{atanh}{\left(\frac{x - 1}{\sqrt{1 + \sqrt{5}}} \right)}}{\sqrt{1 + \sqrt{5}}} & \text{for}\: \left(x - 1\right)^{2} < 1 + \sqrt{5} \end{cases}$$
Gráfica
Respuesta
[src]
______________ / / ______________ \\ ______________ / ______________ \ ______________ / / ______________ \\ ______________ / ______________ \
/ ___ | | / ___ || / ___ | / ___ | / ___ | | / ___ || / ___ | / ___ |
/ 1 \/ 5 | | / 1 \/ 5 / ___\|| / 1 \/ 5 | / 1 \/ 5 / ___\| / 1 \/ 5 | | / 1 \/ 5 / ___\|| / 1 \/ 5 | / 1 \/ 5 / ___\|
/ - -- + ----- *|pi*I + log|1 - / - -- + ----- *\-2 - 2*\/ 5 /|| + / - -- + ----- *log|- / - -- + ----- *\-2 - 2*\/ 5 /| - / - -- + ----- *|pi*I + log|- / - -- + ----- *\-2 - 2*\/ 5 /|| - / - -- + ----- *log|-1 - / - -- + ----- *\-2 - 2*\/ 5 /|
\/ 16 16 \ \ \/ 16 16 // \/ 16 16 \ \/ 16 16 / \/ 16 16 \ \ \/ 16 16 // \/ 16 16 \ \/ 16 16 /
$$- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \log{\left(-1 - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \log{\left(- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(\log{\left(- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + i \pi\right) + \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(\log{\left(1 - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + i \pi\right)$$
=
=
______________ / / ______________ \\ ______________ / ______________ \ ______________ / / ______________ \\ ______________ / ______________ \
/ ___ | | / ___ || / ___ | / ___ | / ___ | | / ___ || / ___ | / ___ |
/ 1 \/ 5 | | / 1 \/ 5 / ___\|| / 1 \/ 5 | / 1 \/ 5 / ___\| / 1 \/ 5 | | / 1 \/ 5 / ___\|| / 1 \/ 5 | / 1 \/ 5 / ___\|
/ - -- + ----- *|pi*I + log|1 - / - -- + ----- *\-2 - 2*\/ 5 /|| + / - -- + ----- *log|- / - -- + ----- *\-2 - 2*\/ 5 /| - / - -- + ----- *|pi*I + log|- / - -- + ----- *\-2 - 2*\/ 5 /|| - / - -- + ----- *log|-1 - / - -- + ----- *\-2 - 2*\/ 5 /|
\/ 16 16 \ \ \/ 16 16 // \/ 16 16 \ \/ 16 16 / \/ 16 16 \ \ \/ 16 16 // \/ 16 16 \ \/ 16 16 /
$$- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \log{\left(-1 - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \log{\left(- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(\log{\left(- \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + i \pi\right) + \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(\log{\left(1 - \sqrt{- \frac{1}{16} + \frac{\sqrt{5}}{16}} \left(- 2 \sqrt{5} - 2\right) \right)} + i \pi\right)$$
sqrt(-1/16 + sqrt(5)/16)*(pi*i + log(1 - sqrt(-1/16 + sqrt(5)/16)*(-2 - 2*sqrt(5)))) + sqrt(-1/16 + sqrt(5)/16)*log(-sqrt(-1/16 + sqrt(5)/16)*(-2 - 2*sqrt(5))) - sqrt(-1/16 + sqrt(5)/16)*(pi*i + log(-sqrt(-1/16 + sqrt(5)/16)*(-2 - 2*sqrt(5)))) - sqrt(-1/16 + sqrt(5)/16)*log(-1 - sqrt(-1/16 + sqrt(5)/16)*(-2 - 2*sqrt(5)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.