What is the integral of ue^{uv} ?

The integral of ue^{uv} is e^{uv}+C

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integral of ue^{uv}

\int u e^{uv} d v

e^{uv} + C

Solution steps

\int u e^{uv} d v

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

w = uv

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

Simplify

u \frac{1}{u} e^{uv} : e^{uv}

= e^{uv}

Add a constant to the solution

= e^{uv} + C