The integral ${{I}_{n}}$ is defined by \[ I_n = \displaystyle\int\nolimits_{n\pi}^{(n+1)\pi} e^{-x}\

Question

The integral ${{I}_{n}}$ is defined by $$ I_n = \displaystyle\int\nolimits_{n\pi}^{(n+1)\pi} e^{-x}\,|\sin x|\,dx, for \quad n \in \mathbb{N}. $$ Show that $$ I_0 = \tfrac{1}{2}\left( 1 + e^{-\pi} \right). $$ [6] By letting $y = x - n\pi$, show that $$ I_n = e^{-n\pi} I_0. $$ [4] Hence determine the exact value of $\displaystyle\int\nolimits_{0}^{\infty }{{{\mathrm{e}}^{-x}}\left| \sin x \right|}\mathrm{ d}x$. [5]

Accepted Answer

(e^π + 1) / (2(e^π − 1))

Exponential Integral Involving Absolute Sine Function

Exponential Integral Involving Absolute Sine Function

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