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