# Multivariable Integration by parts (basic)

Use integration by parts twice on $\int e^{xy} \cos(y) dy$

i wanna integrate $\cos(y)$ first and derivate $e^{xy}$ since x is a constant of the integration.

we have $e^{xy} \sin(y)$ + $\int xe^{xy} \sin(y) dy$

followed by

$e^{xy} \sin(y)$ + $xe^{xy} \cos(y) dy$ + $\int -x^{2}e^{xy} \cos(y) dy$

This doesnt look right....

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Let $$I=\int e^{xy}\cos ydy$$ Then, as you have calculated, $$I=e^{xy}\sin y+xe^{xy}\cos y-x^2I$$ Therefore, $$I=\frac{e^{xy}\sin y+xe^{xy}\cos y}{1+x^2}$$