I am now asked to integrate the following:

[tex]\int^\infty_{-\infty}xAe^{-\lambda(x-a)^2}dx = A\int^\infty_{-\infty}xe^{-\lambda(x-a)^2}dx[/tex]

Where A, a, and [itex]\lambda[/itex] are positive, real constants.

I first tried to do it by parts but that left me with infinities.

Then I tried setting [itex]u = x - a[/itex] and did some re-arranging, which left me with:

[tex]A\int^\infty_{-\infty}(u+a)e^{{-\lambda}u^2}dx = Aa\sqrt\frac{\pi}{\lambda}[/tex]

Is this correct? Also, why does integrated by parts fail?

[tex]\int^\infty_{-\infty}xAe^{-\lambda(x-a)^2}dx = A\int^\infty_{-\infty}xe^{-\lambda(x-a)^2}dx[/tex]

Where A, a, and [itex]\lambda[/itex] are positive, real constants.

I first tried to do it by parts but that left me with infinities.

Then I tried setting [itex]u = x - a[/itex] and did some re-arranging, which left me with:

[tex]A\int^\infty_{-\infty}(u+a)e^{{-\lambda}u^2}dx = Aa\sqrt\frac{\pi}{\lambda}[/tex]

Is this correct? Also, why does integrated by parts fail?

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