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

The integral of ln((e^x)/(x^2)) is (x^2)/2-2xln(x)+2x+C

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

integral

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

\frac{x ^{2}}{2} - 2 x ln (x) + 2 x + C

Solution steps

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

Simplify

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

= \int x - ln (x^{2}) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x d x - \int ln (x^{2}) d x

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

\int ln (x^{2}) d x = 2 x ln (x) - 2 x

= \frac{x ^{2}}{2} - (2 x ln (x) - 2 x)

Simplify

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

Add a constant to the solution

= \frac{x ^{2}}{2} - 2 x ln (x) + 2 x + C

What is the integral of ln((e^x)/(x^2)) ?
The integral of ln((e^x)/(x^2)) is (x^2)/2-2xln(x)+2x+C