Sr Examen
Integral de (R^2-(H-x)^2)^(1/2) dx
Solución
Ha introducido
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1 / | | _______________ | / 2 2 | \/ r - (h - x) dx | / 0
$$\int\limits_{0}^{1} \sqrt{r^{2} - \left(h - x\right)^{2}}\, dx$$
Integral(sqrt(r^2 - (h - x)^2), (x, 0, 1))
Respuesta (Indefinida)
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// 2 /x - h\ \
|| I*r *acosh|-----| 3 | 2| |
|| \ r / I*(x - h) I*r*(x - h) |(h - x) | |
||- ----------------- + ------------------------- - ----------------------- for |--------| > 1|
|| 2 _______________ _______________ | 2 | |
/ || / 2 / 2 | r | |
| || / (x - h) / (x - h) |
| _______________ || 2*r* / -1 + -------- 2* / -1 + -------- |
| / 2 2 || / 2 / 2 |
| \/ r - (h - x) dx = C + |< \/ r \/ r |
| || |
/ || ______________ |
|| / 2 |
|| / (x - h) |
|| 2 /x - h\ r* / 1 - -------- *(x - h) |
|| r *asin|-----| / 2 |
|| \ r / \/ r |
|| -------------- + ------------------------------ otherwise |
\\ 2 2 /
$$\int \sqrt{r^{2} - \left(h - x\right)^{2}}\, dx = C + \begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{- h + x}{r} \right)}}{2} - \frac{i r \left(- h + x\right)}{2 \sqrt{-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}}} + \frac{i \left(- h + x\right)^{3}}{2 r \sqrt{-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{\left(h - x\right)^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{- h + x}{r} \right)}}{2} + \frac{r \sqrt{1 - \frac{\left(- h + x\right)^{2}}{r^{2}}} \left(- h + x\right)}{2} & \text{otherwise} \end{cases}$$
Respuesta
[src]
1 / | | / 2 3 2 | | I*r 3*I*(x - h) I*(x - h) *(-2*h + 2*x) I*(x - h)*(-2*h + 2*x) (x - h) | |- --------------------- + ------------------------- - ----------------------- + ---------------------- for -------- > 1 | | _______________ _______________ 3/2 3/2 | 2| | | / 2 / 2 / 2\ / 2\ |r | | | / (x - h) / (x - h) 3 | (x - h) | | (x - h) | | | / -1 + -------- 2*r* / -1 + -------- 4*r *|-1 + --------| 4*r*|-1 + --------| | | / 2 / 2 | 2 | | 2 | | | \/ r \/ r \ r / \ r / | | | | ______________ | < / 2 dx | | / (x - h) | | r* / 1 - -------- | | / 2 | | \/ r r (x - h)*(-2*h + 2*x) | | ---------------------- + ---------------------- - ------------------------ otherwise | | 2 ______________ ______________ | | / 2 / 2 | | / (x - h) / (x - h) | | 2* / 1 - -------- 4*r* / 1 - -------- | | / 2 / 2 | \ \/ r \/ r | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}}} + \frac{3 i \left(- h + x\right)^{2}}{2 r \sqrt{-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}}} + \frac{i \left(- 2 h + 2 x\right) \left(- h + x\right)}{4 r \left(-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i \left(- 2 h + 2 x\right) \left(- h + x\right)^{3}}{4 r^{3} \left(-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left(- h + x\right)^{2}}{\left|{r^{2}}\right|} > 1 \\\frac{r \sqrt{1 - \frac{\left(- h + x\right)^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{\left(- h + x\right)^{2}}{r^{2}}}} - \frac{\left(- 2 h + 2 x\right) \left(- h + x\right)}{4 r \sqrt{1 - \frac{\left(- h + x\right)^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dx$$
=
=
1 / | | / 2 3 2 | | I*r 3*I*(x - h) I*(x - h) *(-2*h + 2*x) I*(x - h)*(-2*h + 2*x) (x - h) | |- --------------------- + ------------------------- - ----------------------- + ---------------------- for -------- > 1 | | _______________ _______________ 3/2 3/2 | 2| | | / 2 / 2 / 2\ / 2\ |r | | | / (x - h) / (x - h) 3 | (x - h) | | (x - h) | | | / -1 + -------- 2*r* / -1 + -------- 4*r *|-1 + --------| 4*r*|-1 + --------| | | / 2 / 2 | 2 | | 2 | | | \/ r \/ r \ r / \ r / | | | | ______________ | < / 2 dx | | / (x - h) | | r* / 1 - -------- | | / 2 | | \/ r r (x - h)*(-2*h + 2*x) | | ---------------------- + ---------------------- - ------------------------ otherwise | | 2 ______________ ______________ | | / 2 / 2 | | / (x - h) / (x - h) | | 2* / 1 - -------- 4*r* / 1 - -------- | | / 2 / 2 | \ \/ r \/ r | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}}} + \frac{3 i \left(- h + x\right)^{2}}{2 r \sqrt{-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}}} + \frac{i \left(- 2 h + 2 x\right) \left(- h + x\right)}{4 r \left(-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i \left(- 2 h + 2 x\right) \left(- h + x\right)^{3}}{4 r^{3} \left(-1 + \frac{\left(- h + x\right)^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left(- h + x\right)^{2}}{\left|{r^{2}}\right|} > 1 \\\frac{r \sqrt{1 - \frac{\left(- h + x\right)^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{\left(- h + x\right)^{2}}{r^{2}}}} - \frac{\left(- 2 h + 2 x\right) \left(- h + x\right)}{4 r \sqrt{1 - \frac{\left(- h + x\right)^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-i*r/sqrt(-1 + (x - h)^2/r^2) + 3*i*(x - h)^2/(2*r*sqrt(-1 + (x - h)^2/r^2)) - i*(x - h)^3*(-2*h + 2*x)/(4*r^3*(-1 + (x - h)^2/r^2)^(3/2)) + i*(x - h)*(-2*h + 2*x)/(4*r*(-1 + (x - h)^2/r^2)^(3/2)), (x - h)^2/|r^2| > 1), (r*sqrt(1 - (x - h)^2/r^2)/2 + r/(2*sqrt(1 - (x - h)^2/r^2)) - (x - h)*(-2*h + 2*x)/(4*r*sqrt(1 - (x - h)^2/r^2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.