Sr Examen
Integral de exp(-a*x^2)*cos(b*x^2) dx
Solución
Ha introducido
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oo / | | 2 | -a*x / 2\ | e *cos\b*x / dx | / 0
$$\int\limits_{0}^{\infty} e^{- a x^{2}} \cos{\left(b x^{2} \right)}\, dx$$
Integral(exp((-a)*x^2)*cos(b*x^2), (x, 0, oo))
Respuesta
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/ / / ____\\ | | | / 2 || | | | / b || | |atan| / -- || | | | / 2 || | ___ | \\/ a /| |\/ 2 *cos|----------------|*Gamma(1/4)*Gamma(3/4) / / 2 \\ | \ 2 / | | a || |------------------------------------------------- for Or|And(2*|arg(b)| = 0, 2*|arg(a)| < pi), And|2*|arg(a)| = pi, 2*|arg(b)| = 0, -- != -1|| | ________ | | 2 || | / 2 \ \ b // | ____ ___ / b < 4*\/ pi *\/ a * / 1 + -- | 4 / 2 | \/ a | | oo | / | | | | 2 | | / 2\ -a*x | | cos\b*x /*e dx otherwise | | | / \ 0
$$\begin{cases} \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{\frac{b^{2}}{a^{2}}} \right)}}{2} \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)}{4 \sqrt{\pi} \sqrt{a} \sqrt[4]{1 + \frac{b^{2}}{a^{2}}}} & \text{for}\: \left(2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(a \right)}}\right| = \pi \wedge 2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge \frac{a^{2}}{b^{2}} \neq -1\right) \\\int\limits_{0}^{\infty} e^{- a x^{2}} \cos{\left(b x^{2} \right)}\, dx & \text{otherwise} \end{cases}$$
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/ / / ____\\ | | | / 2 || | | | / b || | |atan| / -- || | | | / 2 || | ___ | \\/ a /| |\/ 2 *cos|----------------|*Gamma(1/4)*Gamma(3/4) / / 2 \\ | \ 2 / | | a || |------------------------------------------------- for Or|And(2*|arg(b)| = 0, 2*|arg(a)| < pi), And|2*|arg(a)| = pi, 2*|arg(b)| = 0, -- != -1|| | ________ | | 2 || | / 2 \ \ b // | ____ ___ / b < 4*\/ pi *\/ a * / 1 + -- | 4 / 2 | \/ a | | oo | / | | | | 2 | | / 2\ -a*x | | cos\b*x /*e dx otherwise | | | / \ 0
$$\begin{cases} \frac{\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{\frac{b^{2}}{a^{2}}} \right)}}{2} \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)}{4 \sqrt{\pi} \sqrt{a} \sqrt[4]{1 + \frac{b^{2}}{a^{2}}}} & \text{for}\: \left(2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(a \right)}}\right| = \pi \wedge 2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge \frac{a^{2}}{b^{2}} \neq -1\right) \\\int\limits_{0}^{\infty} e^{- a x^{2}} \cos{\left(b x^{2} \right)}\, dx & \text{otherwise} \end{cases}$$
Piecewise((sqrt(2)*cos(atan(sqrt(b^2/a^2))/2)*gamma(1/4)*gamma(3/4)/(4*sqrt(pi)*sqrt(a)*(1 + b^2/a^2)^(1/4)), ((2*Abs(arg(b)) = 0))∧(2*Abs(arg(a)) < pi))∨((Eq(2*Abs(arg(a), pi))∧(2*Abs(arg(b)) = 0)∧(Ne(a^2/b^2, -1)))), (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.