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

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

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integral of (5x^2-8x-8)/(5x)

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

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

Solution steps

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

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{5 x ^{2} - 8 x - 8}{x} d x

Expand

\frac{5 x ^{2} - 8 x - 8}{x} : 5 x - 8 - \frac{8}{x}

= \frac{1}{5} \cdot \int 5 x - 8 - \frac{8}{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

= \frac{1}{5} (\int 5 x d x - \int 8 d x - \int \frac{8}{x} d x)

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

\int 8 d x = 8 x

\int \frac{8}{x} d x = 8 ln ∣ x ∣

= \frac{1}{5} (\frac{5 x ^{2}}{2} - 8 x - 8 ln ∣ x ∣)

Simplify

\frac{1}{5} (\frac{5 x ^{2}}{2} - 8 x - 8 ln ∣ x ∣) : \frac{x ^{2}}{2} - \frac{8}{5} x - \frac{8}{5} ln ∣ x ∣

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

Add a constant to the solution

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

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