What is the integral of 1/a e^{ax} ?

The integral of 1/a e^{ax} is e^{ax} 1/(a^2)+C

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integral of 1/a e^{ax}

\int \frac{1}{a} e^{a x} d x

e^{a x} \frac{1}{a ^{2}} + C

Solution steps

\int \frac{1}{a} e^{a x} d x

Take the constant out:

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

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

Apply u-substitution

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

= \frac{1}{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

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

Use the common integral:

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

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

Substitute back

u = a x

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

Simplify

\frac{1}{a} \cdot \frac{1}{a} e^{a x} : e^{a x} \frac{1}{a ^{2}}

= e^{a x} \frac{1}{a ^{2}}

Add a constant to the solution

= e^{a x} \frac{1}{a ^{2}} + C

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What is the integral of 1/a e^{ax} ?
The integral of 1/a e^{ax} is e^{ax} 1/(a^2)+C