Sr Examen
Integral de 1/((-x^(-2/3)+c))^1/2 dx
Solución
Ha introducido
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1 / | | 1 | ----------------- dx | ____________ | / 1 | / - ---- + c | / 2/3 | \/ x | / 0
$$\int\limits_{0}^{1} \frac{1}{\sqrt{c - \frac{1}{x^{\frac{2}{3}}}}}\, dx$$
Integral(1/(sqrt(-1/x^(2/3) + c)), (x, 0, 1))
Respuesta
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1 / | | / _____________ | | 3 ___ / 2/3 | | \/ x 2*\/ -1 + c*x 2 2/3 | |------------------ + ------------------ + -------------------------- for x *|c| > 1 | | _____________ 3 ___ _____________ | | / 2/3 3*c*\/ x 3 ___ / 2/3 | |3*\/ -1 + c*x 3*c*\/ x *\/ -1 + c*x | < dx | | ____________ | | 3 ___ / 2/3 | | I*\/ x 2*I 2*I*\/ 1 - c*x | |- ----------------- - ------------------------- + ------------------- otherwise | | ____________ ____________ 3 ___ | | / 2/3 3 ___ / 2/3 3*c*\/ x | \ 3*\/ 1 - c*x 3*c*\/ x *\/ 1 - c*x | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{\sqrt[3]{x}}{3 \sqrt{c x^{\frac{2}{3}} - 1}} + \frac{2 \sqrt{c x^{\frac{2}{3}} - 1}}{3 c \sqrt[3]{x}} + \frac{2}{3 c \sqrt[3]{x} \sqrt{c x^{\frac{2}{3}} - 1}} & \text{for}\: x^{\frac{2}{3}} \left|{c}\right| > 1 \\- \frac{i \sqrt[3]{x}}{3 \sqrt{- c x^{\frac{2}{3}} + 1}} + \frac{2 i \sqrt{- c x^{\frac{2}{3}} + 1}}{3 c \sqrt[3]{x}} - \frac{2 i}{3 c \sqrt[3]{x} \sqrt{- c x^{\frac{2}{3}} + 1}} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / _____________ | | 3 ___ / 2/3 | | \/ x 2*\/ -1 + c*x 2 2/3 | |------------------ + ------------------ + -------------------------- for x *|c| > 1 | | _____________ 3 ___ _____________ | | / 2/3 3*c*\/ x 3 ___ / 2/3 | |3*\/ -1 + c*x 3*c*\/ x *\/ -1 + c*x | < dx | | ____________ | | 3 ___ / 2/3 | | I*\/ x 2*I 2*I*\/ 1 - c*x | |- ----------------- - ------------------------- + ------------------- otherwise | | ____________ ____________ 3 ___ | | / 2/3 3 ___ / 2/3 3*c*\/ x | \ 3*\/ 1 - c*x 3*c*\/ x *\/ 1 - c*x | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{\sqrt[3]{x}}{3 \sqrt{c x^{\frac{2}{3}} - 1}} + \frac{2 \sqrt{c x^{\frac{2}{3}} - 1}}{3 c \sqrt[3]{x}} + \frac{2}{3 c \sqrt[3]{x} \sqrt{c x^{\frac{2}{3}} - 1}} & \text{for}\: x^{\frac{2}{3}} \left|{c}\right| > 1 \\- \frac{i \sqrt[3]{x}}{3 \sqrt{- c x^{\frac{2}{3}} + 1}} + \frac{2 i \sqrt{- c x^{\frac{2}{3}} + 1}}{3 c \sqrt[3]{x}} - \frac{2 i}{3 c \sqrt[3]{x} \sqrt{- c x^{\frac{2}{3}} + 1}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((x^(1/3)/(3*sqrt(-1 + c*x^(2/3))) + 2*sqrt(-1 + c*x^(2/3))/(3*c*x^(1/3)) + 2/(3*c*x^(1/3)*sqrt(-1 + c*x^(2/3))), x^(2/3)*|c| > 1), (-i*x^(1/3)/(3*sqrt(1 - c*x^(2/3))) - 2*i/(3*c*x^(1/3)*sqrt(1 - c*x^(2/3))) + 2*i*sqrt(1 - c*x^(2/3))/(3*c*x^(1/3)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.