how to evaluate this integral: $$l(y)=\int\limits_\beta^\infty \theta\exp(-y\theta)\alpha\exp(-\alpha\theta) \, d\theta$$ where $\alpha,\beta,\theta,y>0.$ Because I find it infinity! Can anyone help me to evaluate this integral? Thank you.$$$$ I find this solution :$$\left(\left.\frac{-1}{(\alpha+y)^2}\exp(-(\alpha+y)\theta)\right)\right|_{\beta}^\infty-\left.\frac{\theta}{(\alpha+y)}\exp(-(\alpha+y)\theta)\right|_{\beta}^\infty.$$ $$$$ in which in the second term, I obtain infinity value!
how can I evaluate this integral?
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integration
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0Try substituting $t = exp(-\alpha \theta)$ . – 2012-11-13
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5Hint: Note that $\ell(y) = - \dfrac{\partial}{\partial y} \int_\beta^\infty \alpha \exp(-(y+\alpha) \theta)\ d\theta$ – 2012-11-13
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0@Dilawar Or not. – 2012-11-13