What is the integral of (x^2+1)/(3x^2-x-2) ?

The integral of (x^2+1)/(3x^2-x-2) is 1/3 x-13/45 ln|3x+2|+2/5 ln|x-1|+C

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integral of (x^2+1)/(3x^2-x-2)

\int \frac{x ^{2} + 1}{3 x ^{2} - x - 2} d x

\frac{1}{3} x - \frac{13}{45} ln ∣ 3 x + 2 ∣ + \frac{2}{5} ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{x ^{2} + 1}{3 x ^{2} - x - 2} d x

Take the partial fraction of

\frac{x ^{2} + 1}{3 x ^{2} - x - 2} : \frac{1}{3} - \frac{13}{15 ( 3 x + 2 )} + \frac{2}{5 ( x - 1 )}

= \int \frac{1}{3} - \frac{13}{15 ( 3 x + 2 )} + \frac{2}{5 ( x - 1 )} 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}{3} d x - \int \frac{13}{15 ( 3 x + 2 )} d x + \int \frac{2}{5 ( x - 1 )} d x

\int \frac{1}{3} d x = \frac{1}{3} x

\int \frac{13}{15 ( 3 x + 2 )} d x = \frac{13}{45} ln ∣ 3 x + 2 ∣

\int \frac{2}{5 ( x - 1 )} d x = \frac{2}{5} ln ∣ x - 1 ∣

= \frac{1}{3} x - \frac{13}{45} ln ∣ 3 x + 2 ∣ + \frac{2}{5} ln ∣ x - 1 ∣

Add a constant to the solution

= \frac{1}{3} x - \frac{13}{45} ln ∣ 3 x + 2 ∣ + \frac{2}{5} ln ∣ x - 1 ∣ + C

t \to 0 + lim (t (ln (t))^{2})

\frac{\partial}{\partial x} (x^{3} + 3 x^{2} y)

x^{2} \frac{d y}{d x} = 4 x^{2} + 7 x y + 2 y^{2}

f (x) = x + 2, at x = 7

\int_{0}^{0.2} 600 x d x

What is the integral of (x^2+1)/(3x^2-x-2) ?
The integral of (x^2+1)/(3x^2-x-2) is 1/3 x-13/45 ln|3x+2|+2/5 ln|x-1|+C