I am trying to integrate the following:
$\int e^{-2y}\cos(y^2) \, dy$
I cannot identify a suitable substitution, and integration by parts would seem to go round in circles.
Please provide me with any minor hints in the right direction (homework).
I am trying to integrate the following:
$\int e^{-2y}\cos(y^2) \, dy$
I cannot identify a suitable substitution, and integration by parts would seem to go round in circles.
Please provide me with any minor hints in the right direction (homework).
Is it given as indefinite integral as you wrote? If it is a definite (improper) integral given from $-\infty$ to $\infty$, I would try by writing $e^{iy^{2}} = \cos(y^{2}) + i\sin(y^{2})$. Convergence seems easy to show in this case.
I don't know if it is doable for indefinite integral.
Notice $\int{e^{-2y}cos(y^2)}dy = -1/2\int{cos(y^2)de^{-2y}}$. Now use Integration by parts. It is not suppose to go round on circles. Just Let $I = \int{e^{-2y}cos(y^2)}dy$, and then until you get the same expression back, you just solve for $I$.