Sr Examen
Integral de sqrt(1-4*x^2)/x dx
Solución
Ha introducido
[src]
sinh(2)
/
|
| __________
| / 2
| \/ 1 - 4*x
| ------------- dx
| x
|
/
sinh(1)
$$\int\limits_{\sinh{\left(1 \right)}}^{\sinh{\left(2 \right)}} \frac{\sqrt{1 - 4 x^{2}}}{x}\, dx$$
Integral(sqrt(1 - 4*x^2)/x, (x, sinh(1), sinh(2)))
Respuesta (Indefinida)
[src]
// / 1 \ 1 2*x 1 \
||- acosh|---| + -------------------- - ---------------- for ------ > 1|
/ || \2*x/ ___________ ___________ | 2| |
| || / 1 / 1 4*|x | |
| __________ || 2*x* / -1 + ---- / -1 + ---- |
| / 2 || / 2 / 2 |
| \/ 1 - 4*x || \/ 4*x \/ 4*x |
| ------------- dx = C + |< |
| x || / 1 \ 2*I*x I |
| || I*asin|---| + --------------- - ------------------- otherwise |
/ || \2*x/ __________ __________ |
|| / 1 / 1 |
|| / 1 - ---- 2*x* / 1 - ---- |
|| / 2 / 2 |
\\ \/ 4*x \/ 4*x /
$$\int \frac{\sqrt{1 - 4 x^{2}}}{x}\, dx = C + \begin{cases} - \frac{2 x}{\sqrt{-1 + \frac{1}{4 x^{2}}}} - \operatorname{acosh}{\left(\frac{1}{2 x} \right)} + \frac{1}{2 x \sqrt{-1 + \frac{1}{4 x^{2}}}} & \text{for}\: \frac{1}{4 \left|{x^{2}}\right|} > 1 \\\frac{2 i x}{\sqrt{1 - \frac{1}{4 x^{2}}}} + i \operatorname{asin}{\left(\frac{1}{2 x} \right)} - \frac{i}{2 x \sqrt{1 - \frac{1}{4 x^{2}}}} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
[src]
2 2
I / 1 \ I / 1 \ 4*I*sinh (1) 4*I*sinh (2)
-------------------- + I*asin|---------| - -------------------- - I*asin|---------| - -------------------- + --------------------
_________________ \2*sinh(2)/ _________________ \2*sinh(1)/ _________________ _________________
/ 2 / 2 / 2 / 2
\/ -1 + 4*sinh (1) \/ -1 + 4*sinh (2) \/ -1 + 4*sinh (1) \/ -1 + 4*sinh (2)
$$- \frac{4 i \sinh^{2}{\left(1 \right)}}{\sqrt{-1 + 4 \sinh^{2}{\left(1 \right)}}} - i \operatorname{asin}{\left(\frac{1}{2 \sinh{\left(1 \right)}} \right)} - \frac{i}{\sqrt{-1 + 4 \sinh^{2}{\left(2 \right)}}} + i \operatorname{asin}{\left(\frac{1}{2 \sinh{\left(2 \right)}} \right)} + \frac{i}{\sqrt{-1 + 4 \sinh^{2}{\left(1 \right)}}} + \frac{4 i \sinh^{2}{\left(2 \right)}}{\sqrt{-1 + 4 \sinh^{2}{\left(2 \right)}}}$$
=
=
2 2
I / 1 \ I / 1 \ 4*I*sinh (1) 4*I*sinh (2)
-------------------- + I*asin|---------| - -------------------- - I*asin|---------| - -------------------- + --------------------
_________________ \2*sinh(2)/ _________________ \2*sinh(1)/ _________________ _________________
/ 2 / 2 / 2 / 2
\/ -1 + 4*sinh (1) \/ -1 + 4*sinh (2) \/ -1 + 4*sinh (1) \/ -1 + 4*sinh (2)
$$- \frac{4 i \sinh^{2}{\left(1 \right)}}{\sqrt{-1 + 4 \sinh^{2}{\left(1 \right)}}} - i \operatorname{asin}{\left(\frac{1}{2 \sinh{\left(1 \right)}} \right)} - \frac{i}{\sqrt{-1 + 4 \sinh^{2}{\left(2 \right)}}} + i \operatorname{asin}{\left(\frac{1}{2 \sinh{\left(2 \right)}} \right)} + \frac{i}{\sqrt{-1 + 4 \sinh^{2}{\left(1 \right)}}} + \frac{4 i \sinh^{2}{\left(2 \right)}}{\sqrt{-1 + 4 \sinh^{2}{\left(2 \right)}}}$$
i/sqrt(-1 + 4*sinh(1)^2) + i*asin(1/(2*sinh(2))) - i/sqrt(-1 + 4*sinh(2)^2) - i*asin(1/(2*sinh(1))) - 4*i*sinh(1)^2/sqrt(-1 + 4*sinh(1)^2) + 4*i*sinh(2)^2/sqrt(-1 + 4*sinh(2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.