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