Sr Examen
Integral de sqrt(x)(x+1)^(-3) dx
Solución
Ha introducido
[src]
1 ---- 5/6 a / | | ___ | \/ x | -------- dx | 3 | (x + 1) | / 5/6 a ---- 4
$$\int\limits_{\frac{a^{\frac{5}{6}}}{4}}^{\frac{1}{a^{\frac{5}{6}}}} \frac{\sqrt{x}}{\left(x + 1\right)^{3}}\, dx$$
Integral(sqrt(x)/(x + 1)^3, (x, a^(5/6)/4, a^(-5/6)))
Respuesta (Indefinida)
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/ | | ___ / ___\ ___ | \/ x atan\\/ x / \/ x *(1 - x) | -------- dx = C + ----------- - ------------- | 3 4 2 | (x + 1) 4*(1 + x) | /
$$\int \frac{\sqrt{x}}{\left(x + 1\right)^{3}}\, dx = C - \frac{\sqrt{x} \left(1 - x\right)}{4 \left(x + 1\right)^{2}} + \frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{4}$$
Respuesta
[src]
/ 1 \ / 5/12\ / 1 \ / 1 \ / 5/12\ / 5/12\
atan|-----| |a | atan|-----| 2*atan|-----| 5/6 |a | 5/3 |a |
| 5/12| 5/12 atan|-----| 5/4 | 5/12| | 5/12| a *atan|-----| a *atan|-----|
1 \a / a 1 \ 2 / a \a / \a / \ 2 / \ 2 /
---------------------- + --------------- + --------------------- - ----------------------- - ----------------- - --------------------- + ---------------------- + ---------------------- - --------------------- - ----------------------
5/4 / 4 8 \ 4 8 / 5/3\ 5/12 / 4 8 \ 5/3 / 5/3\ 5/3 / 4 8 \ 5/6 / 4 8 \ / 5/3\ / 5/3\
a *|4 + ---- + ----| 4 + ---- + ---- | 5/6 a | a *|4 + ---- + ----| 5/6 a | 5/6 a | a *|4 + ---- + ----| a *|4 + ---- + ----| | 5/6 a | | 5/6 a |
| 5/3 5/6| 5/3 5/6 2*|4 + 2*a + ----| | 5/3 5/6| 4 + 2*a + ---- 8*|4 + 2*a + ----| | 5/3 5/6| | 5/3 5/6| 2*|4 + 2*a + ----| 16*|4 + 2*a + ----|
\ a a / a a \ 4 / \ a a / 4 \ 4 / \ a a / \ a a / \ 4 / \ 4 /
$$\frac{a^{\frac{5}{12}}}{2 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{6}} \operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{2 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{4}}}{8 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{3}} \operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{16 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{\operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4} + \frac{\operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}} + \frac{\operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{a^{\frac{5}{3}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} + \frac{1}{a^{\frac{5}{4}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} + \frac{2 \operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{a^{\frac{5}{6}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} - \frac{1}{a^{\frac{5}{12}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)}$$
=
=
/ 1 \ / 5/12\ / 1 \ / 1 \ / 5/12\ / 5/12\
atan|-----| |a | atan|-----| 2*atan|-----| 5/6 |a | 5/3 |a |
| 5/12| 5/12 atan|-----| 5/4 | 5/12| | 5/12| a *atan|-----| a *atan|-----|
1 \a / a 1 \ 2 / a \a / \a / \ 2 / \ 2 /
---------------------- + --------------- + --------------------- - ----------------------- - ----------------- - --------------------- + ---------------------- + ---------------------- - --------------------- - ----------------------
5/4 / 4 8 \ 4 8 / 5/3\ 5/12 / 4 8 \ 5/3 / 5/3\ 5/3 / 4 8 \ 5/6 / 4 8 \ / 5/3\ / 5/3\
a *|4 + ---- + ----| 4 + ---- + ---- | 5/6 a | a *|4 + ---- + ----| 5/6 a | 5/6 a | a *|4 + ---- + ----| a *|4 + ---- + ----| | 5/6 a | | 5/6 a |
| 5/3 5/6| 5/3 5/6 2*|4 + 2*a + ----| | 5/3 5/6| 4 + 2*a + ---- 8*|4 + 2*a + ----| | 5/3 5/6| | 5/3 5/6| 2*|4 + 2*a + ----| 16*|4 + 2*a + ----|
\ a a / a a \ 4 / \ a a / 4 \ 4 / \ a a / \ a a / \ 4 / \ 4 /
$$\frac{a^{\frac{5}{12}}}{2 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{6}} \operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{2 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{4}}}{8 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{a^{\frac{5}{3}} \operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{16 \left(2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4\right)} - \frac{\operatorname{atan}{\left(\frac{a^{\frac{5}{12}}}{2} \right)}}{2 a^{\frac{5}{6}} + \frac{a^{\frac{5}{3}}}{4} + 4} + \frac{\operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}} + \frac{\operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{a^{\frac{5}{3}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} + \frac{1}{a^{\frac{5}{4}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} + \frac{2 \operatorname{atan}{\left(\frac{1}{a^{\frac{5}{12}}} \right)}}{a^{\frac{5}{6}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)} - \frac{1}{a^{\frac{5}{12}} \left(4 + \frac{4}{a^{\frac{5}{3}}} + \frac{8}{a^{\frac{5}{6}}}\right)}$$
1/(a^(5/4)*(4 + 4/a^(5/3) + 8/a^(5/6))) + atan(a^(-5/12))/(4 + 4/a^(5/3) + 8/a^(5/6)) + a^(5/12)/(2*(4 + 2*a^(5/6) + a^(5/3)/4)) - 1/(a^(5/12)*(4 + 4/a^(5/3) + 8/a^(5/6))) - atan(a^(5/12)/2)/(4 + 2*a^(5/6) + a^(5/3)/4) - a^(5/4)/(8*(4 + 2*a^(5/6) + a^(5/3)/4)) + atan(a^(-5/12))/(a^(5/3)*(4 + 4/a^(5/3) + 8/a^(5/6))) + 2*atan(a^(-5/12))/(a^(5/6)*(4 + 4/a^(5/3) + 8/a^(5/6))) - a^(5/6)*atan(a^(5/12)/2)/(2*(4 + 2*a^(5/6) + a^(5/3)/4)) - a^(5/3)*atan(a^(5/12)/2)/(16*(4 + 2*a^(5/6) + a^(5/3)/4))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.