"Compute the following integral: ∫0∞exsin⁡(x)xdx" was answered by students like you

oliviayychengwh

Answered question

2022-01-07

Compute the following integral:

\int_{0}^{\infty} \frac{e^{x} \sin (x)}{x} d x

Answer & Explanation

Mollie Nash

Beginner2022-01-08Added 33 answers

Using Laplace Transform,

L (\sin (x)) = \frac{1}{s^{2} + 1}

L (\frac{\sin (x)}{x}) = \int_{r}^{\infty} \frac{1}{s^{2} + 1} d s = \frac{π}{2} - \arctan (r)

Therefore,

\int_{0}^{\infty} e^{- r x} \frac{\sin (x)}{x} d x = \frac{π}{2} - \arctan (r)

Substituting r=1,

\int_{0}^{\infty} e^{- x} \frac{\sin (x)}{x} d x = \frac{π}{4}

Virginia Palmer

Beginner2022-01-09Added 27 answers

Yet a different approach: parametric integration. Let

F (λ) = \int_{0}^{\infty} \frac{e^{- λ x} \sin (x)}{x} d x, λ > 0

Then,

F^{'} (λ) = - \int_{0}^{\infty} e^{- λ x} \sin (x) d x = - \frac{1}{1 + λ^{2}}

Integrating and taking into account that

lim_{λ \to \infty} F (λ) = 0

we have

F (λ) = \frac{π}{2} - \arctan λ

and

\int_{0}^{\infty} \frac{e^{- x} \sin (x)}{x} d x = F (1) = \frac{π}{4}

star233

Skilled2022-01-11Added 403 answers

Another approach:
$\int_{0}^{\infty} d x \frac{e^{- x} \sin (x)}{x} = \int_{0}^{\infty} d x \frac{e^{- x}}{x} \sum_{k = 0}^{\infty} \frac{(- 1)^{k} x^{2 k + 1}}{(2 k + 1)!} = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{(2 k + 1)!} \int_{0}^{\infty} d x x^{2 k} e^{- x} = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{(2 k + 1)!} (2 k)! = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{2 k + 1} = \frac{π}{4}$

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