Sr Examen
Integral de (cos(6*x)-(1/(Pi))*(sin(6*x)))*e^(5*x-a*x) dx
Solución
Ha introducido
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0 / | | / sin(6*x)\ 5*x - a*x | |cos(6*x) - --------|*E dx | \ pi / | / -oo
$$\int\limits_{-\infty}^{0} e^{- a x + 5 x} \left(- \frac{\sin{\left(6 x \right)}}{\pi} + \cos{\left(6 x \right)}\right)\, dx$$
Integral((cos(6*x) - sin(6*x)/pi)*E^(5*x - a*x), (x, -oo, 0))
Respuesta
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/ /5 a\ | 4*|- - -| | 1 \4 4/ |------------------- + ----------------------- for 2*|arg(5 - a)| < pi | / 2\ / 36 \ 2 | | (5 - a) | |1 + --------|*(5 - a) |6*pi*|1 + --------| | 2| | \ 36 / \ (5 - a) / | < 0 | / | | | | / sin(6*x) \ 5*x - a*x | | |- -------- + cos(6*x)|*e dx otherwise | | \ pi / | | | / | -oo \
$$\begin{cases} \frac{1}{6 \pi \left(\frac{\left(5 - a\right)^{2}}{36} + 1\right)} + \frac{4 \left(\frac{5}{4} - \frac{a}{4}\right)}{\left(1 + \frac{36}{\left(5 - a\right)^{2}}\right) \left(5 - a\right)^{2}} & \text{for}\: 2 \left|{\arg{\left(5 - a \right)}}\right| < \pi \\\int\limits_{-\infty}^{0} \left(- \frac{\sin{\left(6 x \right)}}{\pi} + \cos{\left(6 x \right)}\right) e^{- a x + 5 x}\, dx & \text{otherwise} \end{cases}$$
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/ /5 a\ | 4*|- - -| | 1 \4 4/ |------------------- + ----------------------- for 2*|arg(5 - a)| < pi | / 2\ / 36 \ 2 | | (5 - a) | |1 + --------|*(5 - a) |6*pi*|1 + --------| | 2| | \ 36 / \ (5 - a) / | < 0 | / | | | | / sin(6*x) \ 5*x - a*x | | |- -------- + cos(6*x)|*e dx otherwise | | \ pi / | | | / | -oo \
$$\begin{cases} \frac{1}{6 \pi \left(\frac{\left(5 - a\right)^{2}}{36} + 1\right)} + \frac{4 \left(\frac{5}{4} - \frac{a}{4}\right)}{\left(1 + \frac{36}{\left(5 - a\right)^{2}}\right) \left(5 - a\right)^{2}} & \text{for}\: 2 \left|{\arg{\left(5 - a \right)}}\right| < \pi \\\int\limits_{-\infty}^{0} \left(- \frac{\sin{\left(6 x \right)}}{\pi} + \cos{\left(6 x \right)}\right) e^{- a x + 5 x}\, dx & \text{otherwise} \end{cases}$$
Piecewise((1/(6*pi*(1 + (5 - a)^2/36)) + 4*(5/4 - a/4)/((1 + 36/(5 - a)^2)*(5 - a)^2), 2*Abs(arg(5 - a)) < pi), (Integral((-sin(6*x)/pi + cos(6*x))*exp(5*x - a*x), (x, -oo, 0)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.