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

The integral of (8x^2+2)/(2x^3+x) is ln|2x^2+1|+2ln|x|+C

integral of (8x^2+2)/(2x^3+x)

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

ln 2 x^{2} + 1 + 2 ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{8 x ^{2} + 2}{2 x ^{3} + x} : \frac{8 x ^{2}}{2 x ^{3} + x} + \frac{2}{2 x ^{3} + 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 \frac{8 x ^{2}}{2 x ^{3} + x} d x + \int \frac{2}{2 x ^{3} + x} d x

\int \frac{8 x ^{2}}{2 x ^{3} + x} d x = 2 ln 2 x^{2} + 1

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

= 2 ln 2 x^{2} + 1 + 2 ln ∣ x ∣ - ln 2 x^{2} + 1

Simplify

= ln 2 x^{2} + 1 + 2 ln ∣ x ∣

Add a constant to the solution

= ln 2 x^{2} + 1 + 2 ln ∣ x ∣ + C

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What is the integral of (8x^2+2)/(2x^3+x) ?
The integral of (8x^2+2)/(2x^3+x) is ln|2x^2+1|+2ln|x|+C