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

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

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

\int \frac{ln ( e ^{2 x} )}{x ^{2}} d x

2 ln ∣ x ∣ + C

Solution steps

\int \frac{ln ( e ^{2 x} )}{x ^{2}} d x

Simplify

\frac{ln ( e ^{2 x} )}{x ^{2}} : \frac{2}{x}

= \int \frac{2}{x} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{x} d x

Use the common integral:

\int \frac{1}{x} d x = ln (∣ x ∣)

= 2 ln ∣ x ∣

Add a constant to the solution

= 2 ln ∣ x ∣ + C

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