What is the integral of e^{uv} ?

The integral of e^{uv} is 1/v e^{vu}+C

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integral of e^{uv}

\int e^{uv} d u

\frac{1}{v} e^{vu} + C

Solution steps

\int e^{uv} d u

Apply u-substitution

= \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

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

Use the common integral:

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

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

Substitute back

w = vu

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

Add a constant to the solution

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

\frac{d}{d x} (4 x^{5} - 10 x^{4} + 2)

\frac{d}{d x} (\frac{x ^{2} + x + 4}{x + 1})

\int \frac{x ^{5} - 3 x}{x ^{4}} d x

\int 4 y + 2 x - 5 d x

y = x^{2} + 1, x = 0, x = 6