What is the integral of ae^{ax} ?

The integral of ae^{ax} is e^{ax}+C

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integral of ae^{ax}

\int a e^{a x} d x

e^{a x} + C

Solution steps

\int a e^{a x} d x

Take the constant out:

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

= a \cdot \int e^{a x} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = a x

= a \frac{1}{a} e^{a x}

Simplify

a \frac{1}{a} e^{a x} : e^{a x}

= e^{a x}

Add a constant to the solution

= e^{a x} + C

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