What is the integral of (3x^4+x^2*e^x-x)/(x^2) ?

The integral of (3x^4+x^2*e^x-x)/(x^2) is x^3+e^x-ln|x|+C

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integral of (3x^4+x^2*e^x-x)/(x^2)

\int \frac{3 x ^{4} + x ^{2} \cdot e ^{x} - x}{x ^{2}} d x

x^{3} + e^{x} - ln ∣ x ∣ + C

Solution steps

\int \frac{3 x ^{4} + x ^{2} e ^{x} - x}{x ^{2}} d x

Expand

\frac{3 x ^{4} + x ^{2} e ^{x} - x}{x ^{2}} : 3 x^{2} + e^{x} - \frac{1}{x}

= \int 3 x^{2} + e^{x} - \frac{1}{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 3 x^{2} d x + \int e^{x} d x - \int \frac{1}{x} d x

\int 3 x^{2} d x = x^{3}

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

\int \frac{1}{x} d x = ln ∣ x ∣

= x^{3} + e^{x} - ln ∣ x ∣

Add a constant to the solution

= x^{3} + e^{x} - ln ∣ x ∣ + C

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What is the integral of (3x^4+x^2*e^x-x)/(x^2) ?
The integral of (3x^4+x^2*e^x-x)/(x^2) is x^3+e^x-ln|x|+C