What is the integral of (x^3e^x-2x^2)/(x^3) ?

The integral of (x^3e^x-2x^2)/(x^3) is e^x-2ln|x|+C

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integral of (x^3e^x-2x^2)/(x^3)

\int \frac{x ^{3} e ^{x} - 2 x ^{2}}{x ^{3}} d x

e^{x} - 2 ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{3} e ^{x} - 2 x ^{2}}{x ^{3}} d x

Simplify

\frac{x ^{3} e ^{x} - 2 x ^{2}}{x ^{3}} : \frac{e ^{x} x - 2}{x}

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

Expand

\frac{e ^{x} x - 2}{x} : e^{x} - \frac{2}{x}

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

= e^{x} - 2 ln ∣ x ∣

Add a constant to the solution

= e^{x} - 2 ln ∣ x ∣ + C

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f (x) = \frac{1}{x ^{2}}, (- 2, \frac{1}{4})

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What is the integral of (x^3e^x-2x^2)/(x^3) ?
The integral of (x^3e^x-2x^2)/(x^3) is e^x-2ln|x|+C