Sr Examen
Integral de dx/((sqrt25-4x^2)) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | ____ 2 | \/ 25 - 4*x | / 0
$$\int\limits_{0}^{1} \frac{1}{- 4 x^{2} + \sqrt{25}}\, dx$$
Integral(1/(sqrt(25) - 4*x^2), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-4, c=sqrt(25), context=1/(-4*x**2 + sqrt(25)), symbol=x), False), (ArccothRule(a=1, b=-4, c=sqrt(25), context=1/(-4*x**2 + sqrt(25)), symbol=x), x**2 > 5/4), (ArctanhRule(a=1, b=-4, c=sqrt(25), context=1/(-4*x**2 + sqrt(25)), symbol=x), x**2 < 5/4)], context=1/(-4*x**2 + sqrt(25)), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ___\ \
|| ___ |2*x*\/ 5 | |
||\/ 5 *acoth|---------| |
/ || \ 5 / 2 |
| ||---------------------- for x > 5/4|
| 1 || 10 |
| ------------- dx = C + |< |
| ____ 2 || / ___\ |
| \/ 25 - 4*x || ___ |2*x*\/ 5 | |
| ||\/ 5 *atanh|---------| |
/ || \ 5 / 2 |
||---------------------- for x < 5/4|
\\ 10 /
$$\int \frac{1}{- 4 x^{2} + \sqrt{25}}\, dx = C + \begin{cases} \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} x}{5} \right)}}{10} & \text{for}\: x^{2} > \frac{5}{4} \\\frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} x}{5} \right)}}{10} & \text{for}\: x^{2} < \frac{5}{4} \end{cases}$$
Gráfica
Respuesta
[src]
/ / ___\\ / ___\ / / ___\\ / ___\
___ | | \/ 5 || ___ |\/ 5 | ___ | |\/ 5 || ___ | \/ 5 |
\/ 5 *|pi*I + log|-1 + -----|| \/ 5 *log|-----| \/ 5 *|pi*I + log|-----|| \/ 5 *log|1 + -----|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
20 20 20 20
$$- \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} \right)}}{20} + \frac{\sqrt{5} \log{\left(1 + \frac{\sqrt{5}}{2} \right)}}{20} - \frac{\sqrt{5} \left(\log{\left(-1 + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{20} + \frac{\sqrt{5} \left(\log{\left(\frac{\sqrt{5}}{2} \right)} + i \pi\right)}{20}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___\
___ | | \/ 5 || ___ |\/ 5 | ___ | |\/ 5 || ___ | \/ 5 |
\/ 5 *|pi*I + log|-1 + -----|| \/ 5 *log|-----| \/ 5 *|pi*I + log|-----|| \/ 5 *log|1 + -----|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
20 20 20 20
$$- \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} \right)}}{20} + \frac{\sqrt{5} \log{\left(1 + \frac{\sqrt{5}}{2} \right)}}{20} - \frac{\sqrt{5} \left(\log{\left(-1 + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{20} + \frac{\sqrt{5} \left(\log{\left(\frac{\sqrt{5}}{2} \right)} + i \pi\right)}{20}$$
-sqrt(5)*(pi*i + log(-1 + sqrt(5)/2))/20 - sqrt(5)*log(sqrt(5)/2)/20 + sqrt(5)*(pi*i + log(sqrt(5)/2))/20 + sqrt(5)*log(1 + sqrt(5)/2)/20
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.