Sr Examen
Integral de sqrt(1+i*n*x)/x dx
Solución
Ha introducido
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E / | | ___________ | \/ 1 + I*n*x | ------------- dx | x | / 1
$$\int\limits_{1}^{e} \frac{\sqrt{x i n + 1}}{x}\, dx$$
Integral(sqrt(1 + (i*n)*x)/x, (x, 1, E))
Respuesta (Indefinida)
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/ | | ___________ | \/ 1 + I*n*x / ___________\ ___________ / ___________\ | ------------- dx = C - log\1 + \/ 1 + I*n*x / + 2*\/ 1 + I*n*x + log\-1 + \/ 1 + I*n*x / | x | /
$$\int \frac{\sqrt{x i n + 1}}{x}\, dx = C + 2 \sqrt{i n x + 1} + \log{\left(\sqrt{i n x + 1} - 1 \right)} - \log{\left(\sqrt{i n x + 1} + 1 \right)}$$
Respuesta
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E / | | / -pi*I pi*I | | ------ ---- | | 2 4 | | I*n*e I*e | | ----------------------- - ----------------------------- for x*|n| > 1 | | __________________ ___________ | | / -pi*I / pi*I | | / ------ / ---- | | / 2 / 2 | | \/ -1 + n*x*e ___ 3/2 / e | | \/ n *x * / 1 - ----- | < \/ n*x dx | | | | -pi*I -pi*I | | ------ ------ | | 2 2 | |1 n*e n*e | |- - ---------------------- + --------------------------------------------------- otherwise | |x _________________ / _________________\ _________________ | | / -pi*I | / -pi*I | / -pi*I | | / ------ | / ------ | / ------ | | / 2 | / 2 | / 2 | \ \/ 1 - n*x*e \1 + \/ 1 - n*x*e /*\/ 1 - n*x*e | / 1
$$\int\limits_{1}^{e} \begin{cases} \frac{i n e^{- \frac{i \pi}{2}}}{\sqrt{n x e^{- \frac{i \pi}{2}} - 1}} - \frac{i e^{\frac{i \pi}{4}}}{\sqrt{n} x^{\frac{3}{2}} \sqrt{1 - \frac{e^{\frac{i \pi}{2}}}{n x}}} & \text{for}\: x \left|{n}\right| > 1 \\- \frac{n e^{- \frac{i \pi}{2}}}{\sqrt{- n x e^{- \frac{i \pi}{2}} + 1}} + \frac{n e^{- \frac{i \pi}{2}}}{\sqrt{- n x e^{- \frac{i \pi}{2}} + 1} \left(\sqrt{- n x e^{- \frac{i \pi}{2}} + 1} + 1\right)} + \frac{1}{x} & \text{otherwise} \end{cases}\, dx$$
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E / | | / -pi*I pi*I | | ------ ---- | | 2 4 | | I*n*e I*e | | ----------------------- - ----------------------------- for x*|n| > 1 | | __________________ ___________ | | / -pi*I / pi*I | | / ------ / ---- | | / 2 / 2 | | \/ -1 + n*x*e ___ 3/2 / e | | \/ n *x * / 1 - ----- | < \/ n*x dx | | | | -pi*I -pi*I | | ------ ------ | | 2 2 | |1 n*e n*e | |- - ---------------------- + --------------------------------------------------- otherwise | |x _________________ / _________________\ _________________ | | / -pi*I | / -pi*I | / -pi*I | | / ------ | / ------ | / ------ | | / 2 | / 2 | / 2 | \ \/ 1 - n*x*e \1 + \/ 1 - n*x*e /*\/ 1 - n*x*e | / 1
$$\int\limits_{1}^{e} \begin{cases} \frac{i n e^{- \frac{i \pi}{2}}}{\sqrt{n x e^{- \frac{i \pi}{2}} - 1}} - \frac{i e^{\frac{i \pi}{4}}}{\sqrt{n} x^{\frac{3}{2}} \sqrt{1 - \frac{e^{\frac{i \pi}{2}}}{n x}}} & \text{for}\: x \left|{n}\right| > 1 \\- \frac{n e^{- \frac{i \pi}{2}}}{\sqrt{- n x e^{- \frac{i \pi}{2}} + 1}} + \frac{n e^{- \frac{i \pi}{2}}}{\sqrt{- n x e^{- \frac{i \pi}{2}} + 1} \left(\sqrt{- n x e^{- \frac{i \pi}{2}} + 1} + 1\right)} + \frac{1}{x} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((i*n*exp_polar(-pi*i/2)/sqrt(-1 + n*x*exp_polar(-pi*i/2)) - i*exp_polar(pi*i/4)/(sqrt(n)*x^(3/2)*sqrt(1 - exp_polar(pi*i/2)/(n*x))), x*|n| > 1), (1/x - n*exp_polar(-pi*i/2)/sqrt(1 - n*x*exp_polar(-pi*i/2)) + n*exp_polar(-pi*i/2)/((1 + sqrt(1 - n*x*exp_polar(-pi*i/2)))*sqrt(1 - n*x*exp_polar(-pi*i/2))), True)), (x, 1, E))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.