What is the integral of (e^{ln(ax)})/(x^2+7) ?

The integral of (e^{ln(ax)})/(x^2+7) is a/2 ln|x^2+7|+C

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integral of (e^{ln(ax)})/(x^2+7)

\int \frac{e ^{l n (a x)}}{x ^{2} + 7} d x

\frac{a}{2} ln x^{2} + 7 + C

Solution steps

\int \frac{e ^{l n (a x)}}{x ^{2} + 7} d x

Apply u-substitution

: \int \frac{a}{2 u} d u

= \int \frac{a}{2 u} d u

Take the constant out:

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

= \frac{a}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{a}{2} ln ∣ u ∣

Substitute back

u = x^{2} + 7

= \frac{a}{2} ln x^{2} + 7

Add a constant to the solution

= \frac{a}{2} ln x^{2} + 7 + C

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What is the integral of (e^{ln(ax)})/(x^2+7) ?
The integral of (e^{ln(ax)})/(x^2+7) is a/2 ln|x^2+7|+C