Compute the following integral: ∫0∞exsin⁡(x)xdx

oliviayychengwh

oliviayychengwh

Answered

2022-01-07

Compute the following integral:
0exsin(x)xdx

Answer & Explanation

Mollie Nash

Mollie Nash

Expert

2022-01-08Added 33 answers

Using Laplace Transform,
L(sin(x))=1s2+1
L(sin(x)x)=r1s2+1ds=π2arctan(r)
Therefore,
0erxsin(x)xdx=π2arctan(r)
Substituting r=1,
0exsin(x)xdx=π4
Virginia Palmer

Virginia Palmer

Expert

2022-01-09Added 27 answers

Yet a different approach: parametric integration. Let
F(λ)=0eλxsin(x)xdx, λ>0
Then,
F(λ)=0eλxsin(x)dx=11+λ2
Integrating and taking into account that limλF(λ)=0 we have
F(λ)=π2arctanλ
and0exsin(x)xdx=F(1)=π4
star233

star233

Expert

2022-01-11Added 238 answers

Another approach:
0dxexsin(x)x=0dxexxk=0(1)kx2k+1(2k+1)!=k=0(1)k(2k+1)!0dxx2kex=k=0(1)k(2k+1)!(2k)!=k=0(1)k2k+1=π4

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