What is the integral of (5x^4-8x^2+3x-2)/(x^2) ?

The integral of (5x^4-8x^2+3x-2)/(x^2) is (5x^3)/3-8x+3ln|x|+2/x+C

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integral of (5x^4-8x^2+3x-2)/(x^2)

\int \frac{5 x ^{4} - 8 x ^{2} + 3 x - 2}{x ^{2}} d x

\frac{5 x ^{3}}{3} - 8 x + 3 ln ∣ x ∣ + \frac{2}{x} + C

Solution steps

\int \frac{5 x ^{4} - 8 x ^{2} + 3 x - 2}{x ^{2}} d x

Expand

\frac{5 x ^{4} - 8 x ^{2} + 3 x - 2}{x ^{2}} : 5 x^{2} - 8 + \frac{3}{x} - \frac{2}{x ^{2}}

= \int 5 x^{2} - 8 + \frac{3}{x} - \frac{2}{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 5 x^{2} d x - \int 8 d x + \int \frac{3}{x} d x - \int \frac{2}{x ^{2}} d x

\int 5 x^{2} d x = \frac{5 x ^{3}}{3}

\int 8 d x = 8 x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

= \frac{5 x ^{3}}{3} - 8 x + 3 ln ∣ x ∣ - (- \frac{2}{x})

Simplify

= \frac{5 x ^{3}}{3} - 8 x + 3 ln ∣ x ∣ + \frac{2}{x}

Add a constant to the solution

= \frac{5 x ^{3}}{3} - 8 x + 3 ln ∣ x ∣ + \frac{2}{x} + C

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What is the integral of (5x^4-8x^2+3x-2)/(x^2) ?
The integral of (5x^4-8x^2+3x-2)/(x^2) is (5x^3)/3-8x+3ln|x|+2/x+C