What is the integral of (5x^2-6)/(x^2-2x-48) ?

The integral of (5x^2-6)/(x^2-2x-48) is 5x-87/7 ln|x+6|+157/7 ln|x-8|+C

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integral of (5x^2-6)/(x^2-2x-48)

\int \frac{5 x ^{2} - 6}{x ^{2} - 2 x - 48} d x

5 x - \frac{87}{7} ln ∣ x + 6 ∣ + \frac{157}{7} ln ∣ x - 8 ∣ + C

Solution steps

\int \frac{5 x ^{2} - 6}{x ^{2} - 2 x - 48} d x

Take the partial fraction of

\frac{5 x ^{2} - 6}{x ^{2} - 2 x - 48} : 5 - \frac{87}{7 ( x + 6 )} + \frac{157}{7 ( x - 8 )}

= \int 5 - \frac{87}{7 ( x + 6 )} + \frac{157}{7 ( x - 8 )} 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 5 d x - \int \frac{87}{7 ( x + 6 )} d x + \int \frac{157}{7 ( x - 8 )} d x

\int 5 d x = 5 x

\int \frac{87}{7 ( x + 6 )} d x = \frac{87}{7} ln ∣ x + 6 ∣

\int \frac{157}{7 ( x - 8 )} d x = \frac{157}{7} ln ∣ x - 8 ∣

= 5 x - \frac{87}{7} ln ∣ x + 6 ∣ + \frac{157}{7} ln ∣ x - 8 ∣

Add a constant to the solution

= 5 x - \frac{87}{7} ln ∣ x + 6 ∣ + \frac{157}{7} ln ∣ x - 8 ∣ + C

\frac{d}{d t} (12 t^{2} sin (t))

y = 3 a - b^{2} x^{4}

x \to - 2 lim ((x + 2)^{5} (2 x^{2}))

\frac{\partial}{\partial x} (\frac{- x}{y ^{2}})

\frac{d}{d x} (\frac{6}{x ^{7}})

What is the integral of (5x^2-6)/(x^2-2x-48) ?
The integral of (5x^2-6)/(x^2-2x-48) is 5x-87/7 ln|x+6|+157/7 ln|x-8|+C