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

The integral of (e^{xy}) is 1/y e^{yx}+C

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integral of e^(xy) with respect to x

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

\frac{1}{y} e^{y x} + C

Solution steps

\int e^{x y} d x

Apply u-substitution

: \int \frac{e ^{u}}{y} d u

= \int \frac{e ^{u}}{y} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{y} \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= \frac{1}{y} e^{u}

Substitute back

u = y x

= \frac{1}{y} e^{y x}

Add a constant to the solution

= \frac{1}{y} e^{y x} + C

- \frac{d}{d x} (sec^{2} (x))

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