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