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