Sr Examen
Integral de (1-x^(2/3))^(3/2) dx
Solución
Ha introducido
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1 / | | 3/2 | / 2/3\ | \1 - x / dx | / 0
$$\int\limits_{0}^{1} \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}\, dx$$
Integral((1 - x^(2/3))^(3/2), (x, 0, 1))
Respuesta (Indefinida)
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// /3 ___\ 5/3 3 ___ \
|| 3*I*acosh\\/ x / 9*I*x 3*I*x 3*I*\/ x | 2/3| |
/ ||- ---------------- - ---------------- + ---------------- + ---------------- for |x | > 1|
| || 8 ___________ ___________ ___________ |
| 3/2 // / __________\ 3/2 __________ \ // / __________\ __________ \ || / 2/3 / 2/3 / 2/3 |
| / 2/3\ || | / 2/3 | / 2/3\ 3 ___ / 2/3 / 2/3\ | || | / 2/3 | 3 ___ / 2/3 / 2/3\ | || 8*\/ -1 + x 4*\/ -1 + x 8*\/ -1 + x |
| \1 - x / dx = C - 3*| 0)| ||------------------- - --------------------------------- for And(x <= 1, x > 0)| || /3 ___\ 5/3 3 ___ |
/ \\ 16 6 16 / \\ 8 8 / || 3*asin\\/ x / 3*x 3*\/ x 9*x |
|| ------------- - --------------- - --------------- + --------------- otherwise |
|| 8 __________ __________ __________ |
|| / 2/3 / 2/3 / 2/3 |
\\ 4*\/ 1 - x 8*\/ 1 - x 8*\/ 1 - x /
$$\int \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}\, dx = C + 3 \left(\begin{cases} - \frac{\sqrt[3]{x} \sqrt{1 - x^{\frac{2}{3}}} \left(2 x^{\frac{2}{3}} - 1\right)}{8} + \frac{\operatorname{asin}{\left(\sqrt{1 - x^{\frac{2}{3}}} \right)}}{8} & \text{for}\: x \leq 1 \wedge x > 0 \end{cases}\right) - 3 \left(\begin{cases} - \frac{\sqrt[3]{x} \sqrt{1 - x^{\frac{2}{3}}} \left(2 x^{\frac{2}{3}} - 1\right)}{16} - \frac{x \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}}{6} + \frac{\operatorname{asin}{\left(\sqrt{1 - x^{\frac{2}{3}}} \right)}}{16} & \text{for}\: x \leq 1 \wedge x > 0 \end{cases}\right) + \begin{cases} \frac{3 i x^{\frac{5}{3}}}{4 \sqrt{x^{\frac{2}{3}} - 1}} + \frac{3 i \sqrt[3]{x}}{8 \sqrt{x^{\frac{2}{3}} - 1}} - \frac{9 i x}{8 \sqrt{x^{\frac{2}{3}} - 1}} - \frac{3 i \operatorname{acosh}{\left(\sqrt[3]{x} \right)}}{8} & \text{for}\: \left|{x^{\frac{2}{3}}}\right| > 1 \\- \frac{3 x^{\frac{5}{3}}}{4 \sqrt{1 - x^{\frac{2}{3}}}} - \frac{3 \sqrt[3]{x}}{8 \sqrt{1 - x^{\frac{2}{3}}}} + \frac{9 x}{8 \sqrt{1 - x^{\frac{2}{3}}}} + \frac{3 \operatorname{asin}{\left(\sqrt[3]{x} \right)}}{8} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
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1 / | | / 4/3 4/3 2 2/3 2/3 | | 17*I I 11*I*x 7*I*x I*x 17*I*x 55*I*x 2/3 | |- ----------------- - ----------------- - ----------------- - ---------------- + ---------------- + ----------------- + ----------------- for x > 1 | | ___________ 3/2 3/2 ___________ 3/2 3/2 ___________ | | / 2/3 / 2/3\ / 2/3\ / 2/3 / 2/3\ / 2/3\ / 2/3 | | 16*\/ -1 + x 16*\-1 + x / 24*\-1 + x / 6*\/ -1 + x 6*\-1 + x / 48*\-1 + x / 24*\/ -1 + x | < dx | | 2/3 4/3 2 4/3 2/3 | | 1 17 55*x 11*x x 7*x 17*x | | - ---------------- + ---------------- - ---------------- - ---------------- + --------------- + --------------- + ---------------- otherwise | | 3/2 __________ __________ 3/2 3/2 __________ 3/2 | | / 2/3\ / 2/3 / 2/3 / 2/3\ / 2/3\ / 2/3 / 2/3\ | \ 16*\1 - x / 16*\/ 1 - x 24*\/ 1 - x 24*\1 - x / 6*\1 - x / 6*\/ 1 - x 48*\1 - x / | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{7 i x^{\frac{4}{3}}}{6 \sqrt{x^{\frac{2}{3}} - 1}} - \frac{11 i x^{\frac{4}{3}}}{24 \left(x^{\frac{2}{3}} - 1\right)^{\frac{3}{2}}} + \frac{55 i x^{\frac{2}{3}}}{24 \sqrt{x^{\frac{2}{3}} - 1}} + \frac{17 i x^{\frac{2}{3}}}{48 \left(x^{\frac{2}{3}} - 1\right)^{\frac{3}{2}}} + \frac{i x^{2}}{6 \left(x^{\frac{2}{3}} - 1\right)^{\frac{3}{2}}} - \frac{17 i}{16 \sqrt{x^{\frac{2}{3}} - 1}} - \frac{i}{16 \left(x^{\frac{2}{3}} - 1\right)^{\frac{3}{2}}} & \text{for}\: x^{\frac{2}{3}} > 1 \\\frac{7 x^{\frac{4}{3}}}{6 \sqrt{1 - x^{\frac{2}{3}}}} - \frac{11 x^{\frac{4}{3}}}{24 \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}} - \frac{55 x^{\frac{2}{3}}}{24 \sqrt{1 - x^{\frac{2}{3}}}} + \frac{17 x^{\frac{2}{3}}}{48 \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}} + \frac{x^{2}}{6 \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}} + \frac{17}{16 \sqrt{1 - x^{\frac{2}{3}}}} - \frac{1}{16 \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
=
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1 / | | / 4/3 4/3 2 2/3 2/3 | | 17*I I 11*I*x 7*I*x I*x 17*I*x 55*I*x 2/3 | |- ----------------- - ----------------- - ----------------- - ---------------- + ---------------- + ----------------- + ----------------- for x > 1 | | ___________ 3/2 3/2 ___________ 3/2 3/2 ___________ | | / 2/3 / 2/3\ / 2/3\ / 2/3 / 2/3\ / 2/3\ / 2/3 | | 16*\/ -1 + x 16*\-1 + x / 24*\-1 + x / 6*\/ -1 + x 6*\-1 + x / 48*\-1 + x / 24*\/ -1 + x | < dx | | 2/3 4/3 2 4/3 2/3 | | 1 17 55*x 11*x x 7*x 17*x | | - ---------------- + ---------------- - ---------------- - ---------------- + --------------- + --------------- + ---------------- otherwise | | 3/2 __________ __________ 3/2 3/2 __________ 3/2 | | / 2/3\ / 2/3 / 2/3 / 2/3\ / 2/3\ / 2/3 / 2/3\ | \ 16*\1 - x / 16*\/ 1 - x 24*\/ 1 - x 24*\1 - x / 6*\1 - x / 6*\/ 1 - x 48*\1 - x / | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{7 i x^{\frac{4}{3}}}{6 \sqrt{x^{\frac{2}{3}} - 1}} - \frac{11 i x^{\frac{4}{3}}}{24 \left(x^{\frac{2}{3}} - 1\right)^{\frac{3}{2}}} + \frac{55 i x^{\frac{2}{3}}}{24 \sqrt{x^{\frac{2}{3}} - 1}} + \frac{17 i x^{\frac{2}{3}}}{48 \left(x^{\frac{2}{3}} - 1\right)^{\frac{3}{2}}} + \frac{i x^{2}}{6 \left(x^{\frac{2}{3}} - 1\right)^{\frac{3}{2}}} - \frac{17 i}{16 \sqrt{x^{\frac{2}{3}} - 1}} - \frac{i}{16 \left(x^{\frac{2}{3}} - 1\right)^{\frac{3}{2}}} & \text{for}\: x^{\frac{2}{3}} > 1 \\\frac{7 x^{\frac{4}{3}}}{6 \sqrt{1 - x^{\frac{2}{3}}}} - \frac{11 x^{\frac{4}{3}}}{24 \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}} - \frac{55 x^{\frac{2}{3}}}{24 \sqrt{1 - x^{\frac{2}{3}}}} + \frac{17 x^{\frac{2}{3}}}{48 \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}} + \frac{x^{2}}{6 \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}} + \frac{17}{16 \sqrt{1 - x^{\frac{2}{3}}}} - \frac{1}{16 \left(1 - x^{\frac{2}{3}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-17*i/(16*sqrt(-1 + x^(2/3))) - i/(16*(-1 + x^(2/3))^(3/2)) - 11*i*x^(4/3)/(24*(-1 + x^(2/3))^(3/2)) - 7*i*x^(4/3)/(6*sqrt(-1 + x^(2/3))) + i*x^2/(6*(-1 + x^(2/3))^(3/2)) + 17*i*x^(2/3)/(48*(-1 + x^(2/3))^(3/2)) + 55*i*x^(2/3)/(24*sqrt(-1 + x^(2/3))), x^(2/3) > 1), (-1/(16*(1 - x^(2/3))^(3/2)) + 17/(16*sqrt(1 - x^(2/3))) - 55*x^(2/3)/(24*sqrt(1 - x^(2/3))) - 11*x^(4/3)/(24*(1 - x^(2/3))^(3/2)) + x^2/(6*(1 - x^(2/3))^(3/2)) + 7*x^(4/3)/(6*sqrt(1 - x^(2/3))) + 17*x^(2/3)/(48*(1 - x^(2/3))^(3/2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.