What is the integral of e^{u/v} ?

The integral of e^{u/v} is ve^{u/v}+C

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integral of e^{u/v}

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

v e^{\frac{u}{v}} + C

Solution steps

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

Apply u-substitution

: \int e^{w} v d w

= \int e^{w} v d w

Take the constant out:

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

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

Use the common integral:

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

= v e^{w}

Substitute back

w = \frac{u}{v}

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

Add a constant to the solution

= v e^{\frac{u}{v}} + C

\int (\frac{u - 2}{u}) d u

\int \frac{X ^{2}}{2} d X

\int 3 x e^{8 x} d x

x \to 0 lim (\frac{3 x ^{2} - 3 x}{x})

\frac{d}{d x} (e^{x} \cdot x^{2})