What is the integral of (10)/(3x^3+5x) ?

The integral of (10)/(3x^3+5x) is 2ln|x|-ln|3x^2+5|+C

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integral of (10)/(3x^3+5x)

\int \frac{10}{3 x ^{3} + 5 x} d x

2 ln ∣ x ∣ - ln 3 x^{2} + 5 + C

Solution steps

\int \frac{10}{3 x ^{3} + 5 x} d x

Take the constant out:

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

= 10 \cdot \int \frac{1}{3 x ^{3} + 5 x} d x

Take the partial fraction of

\frac{1}{3 x ^{3} + 5 x} : \frac{1}{5 x} - \frac{3 x}{5 ( 3 x ^{2} + 5 )}

= 10 \cdot \int \frac{1}{5 x} - \frac{3 x}{5 ( 3 x ^{2} + 5 )} 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

= 10 (\int \frac{1}{5 x} d x - \int \frac{3 x}{5 ( 3 x ^{2} + 5 )} d x)

\int \frac{1}{5 x} d x = \frac{1}{5} ln ∣ x ∣

\int \frac{3 x}{5 ( 3 x ^{2} + 5 )} d x = \frac{1}{10} ln 3 x^{2} + 5

= 10 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{10} ln 3 x^{2} + 5)

Simplify

10 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{10} ln 3 x^{2} + 5) : 2 ln ∣ x ∣ - ln 3 x^{2} + 5

= 2 ln ∣ x ∣ - ln 3 x^{2} + 5

Add a constant to the solution

= 2 ln ∣ x ∣ - ln 3 x^{2} + 5 + C

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What is the integral of (10)/(3x^3+5x) ?
The integral of (10)/(3x^3+5x) is 2ln|x|-ln|3x^2+5|+C