What is the integral of ln(e^{xy}cos(x)) ?

The integral of ln(e^{xy}cos(x)) is ln(cos(x)e^{xy})y-(xy^2)/2+C

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integral of ln(e^{xy}cos(x))

\int ln (e^{x y} cos (x)) d y

ln (cos (x) e^{x y}) y - \frac{x y ^{2}}{2} + C

Solution steps

\int ln (e^{x y} cos (x)) d y

Apply Integration By Parts

: ln (cos (x) e^{x y}) y - \int x y d y

= ln (cos (x) e^{x y}) y - \int x y d y

\int x y d y = \frac{x y ^{2}}{2}

= ln (cos (x) e^{x y}) y - \frac{x y ^{2}}{2}

Add a constant to the solution

= ln (cos (x) e^{x y}) y - \frac{x y ^{2}}{2} + C

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\int_{0}^{\frac{π}{2}} tan (\frac{x}{2}) d x

25 y^{^{''}} - 40 y^{^{'}} + 16 y = 0, y (0) = 4, y^{^{'}} (0) = - 5

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What is the integral of ln(e^{xy}cos(x)) ?
The integral of ln(e^{xy}cos(x)) is ln(cos(x)e^{xy})y-(xy^2)/2+C