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

The integral of (3x^2)/(x^2+x-2) is 3x+ln|x-1|-4ln|x+2|+C

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integral of (3x^2)/(x^2+x-2)

\int \frac{3 x ^{2}}{x ^{2} + x - 2} d x

3 x + ln ∣ x - 1 ∣ - 4 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{3 x ^{2}}{x ^{2} + x - 2} d x

Take the constant out:

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

= 3 \cdot \int \frac{x ^{2}}{x ^{2} + x - 2} d x

Take the partial fraction of

\frac{x ^{2}}{x ^{2} + x - 2} : 1 + \frac{1}{3 ( x - 1 )} - \frac{4}{3 ( x + 2 )}

= 3 \cdot \int 1 + \frac{1}{3 ( x - 1 )} - \frac{4}{3 ( x + 2 )} 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

= 3 (\int 1 d x + \int \frac{1}{3 ( x - 1 )} d x - \int \frac{4}{3 ( x + 2 )} d x)

\int 1 d x = x

\int \frac{1}{3 ( x - 1 )} d x = \frac{1}{3} ln ∣ x - 1 ∣

\int \frac{4}{3 ( x + 2 )} d x = \frac{4}{3} ln ∣ x + 2 ∣

= 3 (x + \frac{1}{3} ln ∣ x - 1 ∣ - \frac{4}{3} ln ∣ x + 2 ∣)

Simplify

3 (x + \frac{1}{3} ln ∣ x - 1 ∣ - \frac{4}{3} ln ∣ x + 2 ∣) : 3 x + ln ∣ x - 1 ∣ - 4 ln ∣ x + 2 ∣

= 3 x + ln ∣ x - 1 ∣ - 4 ln ∣ x + 2 ∣

Add a constant to the solution

= 3 x + ln ∣ x - 1 ∣ - 4 ln ∣ x + 2 ∣ + C

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What is the integral of (3x^2)/(x^2+x-2) ?
The integral of (3x^2)/(x^2+x-2) is 3x+ln|x-1|-4ln|x+2|+C