Sr Examen
Integral de cos(x*a)/((x^2+b^2)^2) dx
Solución
Ha introducido
[src]
oo / | | cos(x*a) | ---------- dx | 2 | / 2 2\ | \x + b / | / 0
$$\int\limits_{0}^{\infty} \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx$$
Integral(cos(x*a)/(x^2 + b^2)^2, (x, 0, oo))
Respuesta (Indefinida)
[src]
/ / | | | cos(x*a) | cos(x*a) | ---------- dx = C + | ---------- dx | 2 | 2 | / 2 2\ | / 2 2\ | \x + b / | \b + x / | | / /
$$\int \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx = C + \int \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx$$
Respuesta
[src]
/ / ____ \ | ____ | ____ 4 4 / sinh(a*b) cosh(a*b)\ a*b*\/ pi *(-a*b*sinh(a*b) + cosh(a*b))| |\/ pi *|\/ pi *a *b *|- --------- + ---------| + ---------------------------------------| | | | 3 3 2 2 | 4 | | \ \ 4*a *b 4*a *b / / |----------------------------------------------------------------------------------------- for And(2*|arg(a)| = 0, 2*|arg(b)| < pi) | 4 | a*b | | oo < / | | | | cos(a*x) | | ---------- dx otherwise | | 2 | | / 2 2\ | | \b + x / | | | / | 0 \
$$\begin{cases} \frac{\sqrt{\pi} \left(\sqrt{\pi} a^{4} b^{4} \left(\frac{\cosh{\left(a b \right)}}{4 a^{2} b^{2}} - \frac{\sinh{\left(a b \right)}}{4 a^{3} b^{3}}\right) + \frac{\sqrt{\pi} a b \left(- a b \sinh{\left(a b \right)} + \cosh{\left(a b \right)}\right)}{4}\right)}{a b^{4}} & \text{for}\: 2 \left|{\arg{\left(a \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(b \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx & \text{otherwise} \end{cases}$$
=
=
/ / ____ \ | ____ | ____ 4 4 / sinh(a*b) cosh(a*b)\ a*b*\/ pi *(-a*b*sinh(a*b) + cosh(a*b))| |\/ pi *|\/ pi *a *b *|- --------- + ---------| + ---------------------------------------| | | | 3 3 2 2 | 4 | | \ \ 4*a *b 4*a *b / / |----------------------------------------------------------------------------------------- for And(2*|arg(a)| = 0, 2*|arg(b)| < pi) | 4 | a*b | | oo < / | | | | cos(a*x) | | ---------- dx otherwise | | 2 | | / 2 2\ | | \b + x / | | | / | 0 \
$$\begin{cases} \frac{\sqrt{\pi} \left(\sqrt{\pi} a^{4} b^{4} \left(\frac{\cosh{\left(a b \right)}}{4 a^{2} b^{2}} - \frac{\sinh{\left(a b \right)}}{4 a^{3} b^{3}}\right) + \frac{\sqrt{\pi} a b \left(- a b \sinh{\left(a b \right)} + \cosh{\left(a b \right)}\right)}{4}\right)}{a b^{4}} & \text{for}\: 2 \left|{\arg{\left(a \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(b \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{\cos{\left(a x \right)}}{\left(b^{2} + x^{2}\right)^{2}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((sqrt(pi)*(sqrt(pi)*a^4*b^4*(-sinh(a*b)/(4*a^3*b^3) + cosh(a*b)/(4*a^2*b^2)) + a*b*sqrt(pi)*(-a*b*sinh(a*b) + cosh(a*b))/4)/(a*b^4), (2*Abs(arg(a)) = 0))∧(2*Abs(arg(b)) < pi), (Integral(cos(a*x)/(b^2 + x^2)^2, (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.