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

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

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

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

ln ∣ x ∣ + 2 ln x^{2} + 4 + C

Solution steps

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

Take the partial fraction of

\frac{5 x ^{2} + 4}{x ^{3} + 4 x} : \frac{1}{x} + \frac{4 x}{x ^{2} + 4}

= \int \frac{1}{x} + \frac{4 x}{x ^{2} + 4} 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{1}{x} d x + \int \frac{4 x}{x ^{2} + 4} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{4 x}{x ^{2} + 4} d x = 2 ln x^{2} + 4

= ln ∣ x ∣ + 2 ln x^{2} + 4

Add a constant to the solution

= ln ∣ x ∣ + 2 ln x^{2} + 4 + C

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