Sr Examen
Integral de x*exp(-a*x^2)*cos(b*x) dx
Solución
Ha introducido
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oo / | | 2 | -a*x | x*e *cos(b*x) dx | / 0
$$\int\limits_{0}^{\infty} x e^{- a x^{2}} \cos{\left(b x \right)}\, dx$$
Integral((x*exp((-a)*x^2))*cos(b*x), (x, 0, oo))
Respuesta
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/ / 2\ | | ____ / I*b \ b | 2 | | \/ pi *erf|-------| ---| -b | | | ___| ___ 4*a| ---- | | \2*\/ a / I*\/ a *e | 4*a |-2*I*b*|- ------------------- + ------------|*e | \ 8 4*b / / / pi\ pi\ |--------------------------------------------------- for Or|And|2*|arg(b)| = 0, |arg(a)| < --|, |arg(a)| < --| | 3/2 \ \ 2 / 2 / | a < | oo | / | | | | 2 | | -a*x | | x*cos(b*x)*e dx otherwise | | | / | 0 \
$$\begin{cases} - \frac{2 i b \left(\frac{i \sqrt{a} e^{\frac{b^{2}}{4 a}}}{4 b} - \frac{\sqrt{\pi} \operatorname{erf}{\left(\frac{i b}{2 \sqrt{a}} \right)}}{8}\right) e^{- \frac{b^{2}}{4 a}}}{a^{\frac{3}{2}}} & \text{for}\: \left(2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} x e^{- a x^{2}} \cos{\left(b x \right)}\, dx & \text{otherwise} \end{cases}$$
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/ / 2\ | | ____ / I*b \ b | 2 | | \/ pi *erf|-------| ---| -b | | | ___| ___ 4*a| ---- | | \2*\/ a / I*\/ a *e | 4*a |-2*I*b*|- ------------------- + ------------|*e | \ 8 4*b / / / pi\ pi\ |--------------------------------------------------- for Or|And|2*|arg(b)| = 0, |arg(a)| < --|, |arg(a)| < --| | 3/2 \ \ 2 / 2 / | a < | oo | / | | | | 2 | | -a*x | | x*cos(b*x)*e dx otherwise | | | / | 0 \
$$\begin{cases} - \frac{2 i b \left(\frac{i \sqrt{a} e^{\frac{b^{2}}{4 a}}}{4 b} - \frac{\sqrt{\pi} \operatorname{erf}{\left(\frac{i b}{2 \sqrt{a}} \right)}}{8}\right) e^{- \frac{b^{2}}{4 a}}}{a^{\frac{3}{2}}} & \text{for}\: \left(2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} x e^{- a x^{2}} \cos{\left(b x \right)}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-2*i*b*(-sqrt(pi)*erf(i*b/(2*sqrt(a)))/8 + i*sqrt(a)*exp(b^2/(4*a))/(4*b))*exp(-b^2/(4*a))/a^(3/2), (Abs(arg(a)) < pi/2)∨((2*Abs(arg(b)) = 0))∧(Abs(arg(a)) < pi/2)), (Integral(x*cos(b*x)*exp(-a*x^2), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.