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

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

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

\int ln ∣ e^{x y} cos (x) ∣ d y

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

Solution steps

\int ln ∣ e^{x y} cos (x) ∣ d y

Apply Integration By Parts

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

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

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

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

Apply absolute rule:

a > 0, ∣ a^{x} ∣ = a^{x}

∣ e^{x y} ∣ = e^{x y}

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

Add a constant to the solution

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

\int t \cdot ln (t + 1) d t

\frac{d}{d x} (\frac{3}{x + 9})

- 7 z^{4} + z^{2} - 25

\frac{\partial}{\partial x} (\frac{( x ^{2} )}{y + 3})

x \to 4 lim (\frac{∣ x ∣ - 4}{x - 4})

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