What is the integral of e^{ux} ?

The integral of e^{ux} is 1/u e^{ux}+C

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integral of e^{ux}

\int e^{ux} d x

\frac{1}{u} e^{ux} + C

Solution steps

\int e^{ux} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

v = ux

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

Add a constant to the solution

= \frac{1}{u} e^{ux} + C

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