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Question Number 31053 by abdo imad last updated on 02/Mar/18

$$let\lambda \in Randa>0find{\int }_0^{\mathrm{\infty }}{e}^{-ax}cos\left(\lambda x\right)dx.$$

Commented by abdo imad last updated on 03/Mar/18

$$letputI={\int }_0^{\mathrm{\infty }}{e}^{-ax}cos\left(\lambda x\right)dx$$$$I=Re\left({\int }_0^{\mathrm{\infty }}{e}^{-ax}{e}^{i\lambda x}dx\right)=Re\left({\int }_0^{\mathrm{\infty }}{e}^{(-a+i\lambda )x}dx\right)but$$$${\int }_0^{\mathrm{\infty }}{e}^{(-a+i\lambda )x}dx={\left[\frac{1}{-a+i\lambda }{e}^{(-a+i\lambda )x}\right]}_{x=0}^{x\to +\mathrm{\infty }}$$$$=\frac{-1}{-a+i\lambda }=\frac{1}{a-\lambda i}=\frac{a+\lambda i}{a^2+{\lambda }^2}\Rightarrow I=\frac{a}{a^2+{\lambda }^2}.$$

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