What is the integral of (4x^2+6)/(x^3+3x) ?

The integral of (4x^2+6)/(x^3+3x) is ln|x^2+3|+2ln|x|+C

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integral of (4x^2+6)/(x^3+3x)

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

ln x^{2} + 3 + 2 ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{4 x ^{2} + 6}{x ^{3} + 3 x} : \frac{4 x ^{2}}{x ^{3} + 3 x} + \frac{6}{x ^{3} + 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{4 x ^{2}}{x ^{3} + 3 x} d x + \int \frac{6}{x ^{3} + 3 x} d x

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

\int \frac{6}{x ^{3} + 3 x} d x = 2 ln ∣ x ∣ - ln x^{2} + 3

= 2 ln x^{2} + 3 + 2 ln ∣ x ∣ - ln x^{2} + 3

Simplify

= ln x^{2} + 3 + 2 ln ∣ x ∣

Add a constant to the solution

= ln x^{2} + 3 + 2 ln ∣ x ∣ + C

f (x) = 7 x - 1

y^{^{''}} + 4 y^{^{'}} + 6 y = 0

\int_{- 1}^{1} (x^{2} + 1) d x

y = ln (\frac{e ^{x}}{e ^{x} - 1})

- 4 x^{2} - 7 x

What is the integral of (4x^2+6)/(x^3+3x) ?
The integral of (4x^2+6)/(x^3+3x) is ln|x^2+3|+2ln|x|+C