What is the integral of (12x^2+16)/(x^3+4x) ?

The integral of (12x^2+16)/(x^3+4x) is 4ln|x^2+4|+4ln|x|+C

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integral of (12x^2+16)/(x^3+4x)

\int \frac{12 x ^{2} + 16}{x ^{3} + 4 x} d x

4 ln x^{2} + 4 + 4 ln ∣ x ∣ + C

Solution steps

\int \frac{12 x ^{2} + 16}{x ^{3} + 4 x} d x

Expand

\frac{12 x ^{2} + 16}{x ^{3} + 4 x} : \frac{12 x ^{2}}{x ^{3} + 4 x} + \frac{16}{x ^{3} + 4 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{12 x ^{2}}{x ^{3} + 4 x} d x + \int \frac{16}{x ^{3} + 4 x} d x

\int \frac{12 x ^{2}}{x ^{3} + 4 x} d x = 6 ln x^{2} + 4

\int \frac{16}{x ^{3} + 4 x} d x = 16 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

= 6 ln x^{2} + 4 + 16 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

Simplify

6 ln x^{2} + 4 + 16 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4) : 4 ln x^{2} + 4 + 4 ln ∣ x ∣

= 4 ln x^{2} + 4 + 4 ln ∣ x ∣

Add a constant to the solution

= 4 ln x^{2} + 4 + 4 ln ∣ x ∣ + C

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What is the integral of (12x^2+16)/(x^3+4x) ?
The integral of (12x^2+16)/(x^3+4x) is 4ln|x^2+4|+4ln|x|+C