What is the integral of (5x^2+2)/(0.5x^3+0.6x) ?

The integral of (5x^2+2)/(0.5x^3+0.6x) is 3.33333…ln|x|+3.33333…ln|0.5x^2+0.6|+C

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integral of (5x^2+2)/(0.5x^3+0.6x)

\int \frac{5 x ^{2} + 2}{0.5 x ^{3} + 0.6 x} d x

3.33333 \dots ln ∣ x ∣ + 3.33333 \dots ln 0.5 x^{2} + 0.6 + C

Solution steps

\int \frac{5 x ^{2} + 2}{0.5 x ^{3} + 0.6 x} d x

Factor

\frac{5 x ^{2} + 2}{0.5 x ^{3} + 0.6 x}

= \int \frac{5 x ^{2} + 2}{x ( 0.5 x ^{2} + 0.6 )} d x

Take the partial fraction of

\frac{5 x ^{2} + 2}{x ( 0.5 x ^{2} + 0.6 )} : \frac{10}{3 x} + \frac{3.33333 \dots x}{0.5 x ^{2} + 0.6}

= \int \frac{10}{3 x} + \frac{3.33333 \dots x}{0.5 x ^{2} + 0.6} 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 \frac{10}{3 x} d x + \int \frac{3.33333 \dots x}{0.5 x ^{2} + 0.6} d x

\int \frac{10}{3 x} d x = \frac{10}{3} ln ∣ x ∣

\int \frac{3.33333 \dots x}{0.5 x ^{2} + 0.6} d x = 3.33333 \dots ln 0.5 x^{2} + 0.6

= \frac{10}{3} ln ∣ x ∣ + 3.33333 \dots ln 0.5 x^{2} + 0.6

Simplify

= 3.33333 \dots ln ∣ x ∣ + 3.33333 \dots ln 0.5 x^{2} + 0.6

Add a constant to the solution

= 3.33333 \dots ln ∣ x ∣ + 3.33333 \dots ln 0.5 x^{2} + 0.6 + C

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What is the integral of (5x^2+2)/(0.5x^3+0.6x) ?
The integral of (5x^2+2)/(0.5x^3+0.6x) is 3.33333…ln|x|+3.33333…ln|0.5x^2+0.6|+C