What is the integral of e^{ax} ?

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

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integral of e^{ax}

\int e^{a x} d x

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

Solution steps

\int e^{a x} d x

Apply u-substitution

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

= \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 \int e^{u} d u

Use the common integral:

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

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

Substitute back

u = a x

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

Add a constant to the solution

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

f (x) = \frac{1 - x ^{2}}{2 x + 1}

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