Sr Examen
Integral de x^(3/2)*(1+x)^(1/2) dx
Solución
Ha introducido
[src]
1 / | | 3/2 _______ | x *\/ 1 + x dx | / 0
$$\int\limits_{0}^{1} x^{\frac{3}{2}} \sqrt{x + 1}\, dx$$
Integral(x^(3/2)*sqrt(1 + x), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / _______\ 3/2 5/2 _______ \ // / _______\ 5/2 3/2 7/2 _______ \
|| acosh\\/ 1 + x / 3*(1 + x) (1 + x) \/ 1 + x | || acosh\\/ 1 + x / 5*(1 + x) (1 + x) (1 + x) \/ 1 + x |
/ || - ---------------- - ------------ + ---------- + --------- for |1 + x| > 1| || - ---------------- - ------------ - ---------- + ---------- + --------- for |1 + x| > 1|
| || 8 ___ ___ ___ | || 16 ___ ___ ___ ___ |
| 3/2 _______ || 8*\/ x 4*\/ x 8*\/ x | || 24*\/ x 48*\/ x 6*\/ x 16*\/ x |
| x *\/ 1 + x dx = C - 2*|< | + 2*|< |
| || / _______\ 5/2 _______ 3/2 | || / _______\ 7/2 _______ 3/2 5/2 |
/ ||I*asin\\/ 1 + x / I*(1 + x) I*\/ 1 + x 3*I*(1 + x) | ||I*asin\\/ 1 + x / I*(1 + x) I*\/ 1 + x I*(1 + x) 5*I*(1 + x) |
||----------------- - ------------ - ----------- + -------------- otherwise | ||----------------- - ------------ - ----------- + ------------ + -------------- otherwise |
|| 8 ____ ____ ____ | || 16 ____ ____ ____ ____ |
\\ 4*\/ -x 8*\/ -x 8*\/ -x / \\ 6*\/ -x 16*\/ -x 48*\/ -x 24*\/ -x /
$$\int x^{\frac{3}{2}} \sqrt{x + 1}\, dx = C - 2 \left(\begin{cases} - \frac{\operatorname{acosh}{\left(\sqrt{x + 1} \right)}}{8} + \frac{\left(x + 1\right)^{\frac{5}{2}}}{4 \sqrt{x}} - \frac{3 \left(x + 1\right)^{\frac{3}{2}}}{8 \sqrt{x}} + \frac{\sqrt{x + 1}}{8 \sqrt{x}} & \text{for}\: \left|{x + 1}\right| > 1 \\\frac{i \operatorname{asin}{\left(\sqrt{x + 1} \right)}}{8} - \frac{i \left(x + 1\right)^{\frac{5}{2}}}{4 \sqrt{- x}} + \frac{3 i \left(x + 1\right)^{\frac{3}{2}}}{8 \sqrt{- x}} - \frac{i \sqrt{x + 1}}{8 \sqrt{- x}} & \text{otherwise} \end{cases}\right) + 2 \left(\begin{cases} - \frac{\operatorname{acosh}{\left(\sqrt{x + 1} \right)}}{16} + \frac{\left(x + 1\right)^{\frac{7}{2}}}{6 \sqrt{x}} - \frac{5 \left(x + 1\right)^{\frac{5}{2}}}{24 \sqrt{x}} - \frac{\left(x + 1\right)^{\frac{3}{2}}}{48 \sqrt{x}} + \frac{\sqrt{x + 1}}{16 \sqrt{x}} & \text{for}\: \left|{x + 1}\right| > 1 \\\frac{i \operatorname{asin}{\left(\sqrt{x + 1} \right)}}{16} - \frac{i \left(x + 1\right)^{\frac{7}{2}}}{6 \sqrt{- x}} + \frac{5 i \left(x + 1\right)^{\frac{5}{2}}}{24 \sqrt{- x}} + \frac{i \left(x + 1\right)^{\frac{3}{2}}}{48 \sqrt{- x}} - \frac{i \sqrt{x + 1}}{16 \sqrt{- x}} & \text{otherwise} \end{cases}\right)$$
Gráfica
Respuesta
[src]
/ ___\ ___
acosh\\/ 2 / 7*\/ 2
------------ + -------
8 24
$$\frac{\operatorname{acosh}{\left(\sqrt{2} \right)}}{8} + \frac{7 \sqrt{2}}{24}$$
=
=
/ ___\ ___
acosh\\/ 2 / 7*\/ 2
------------ + -------
8 24
$$\frac{\operatorname{acosh}{\left(\sqrt{2} \right)}}{8} + \frac{7 \sqrt{2}}{24}$$
acosh(sqrt(2))/8 + 7*sqrt(2)/24
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.