What is the integral of ve^{uv} ?

The integral of ve^{uv} is e^{vu}+C

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integral of ve^{uv}

\int v e^{uv} d u

e^{vu} + C

Solution steps

\int v e^{uv} d u

Take the constant out:

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

= v \cdot \int e^{vu} d u

Apply u-substitution

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

= v \cdot \int \frac{e ^{w}}{v} d w

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

w = vu

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

Simplify

v \frac{1}{v} e^{vu} : e^{vu}

= e^{vu}

Add a constant to the solution

= e^{vu} + C

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