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

The integral of x/(x^2-3x-4) is 1/5 ln|x+1|+4/5 ln|x-4|+C

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integral of x/(x^2-3x-4)

\int \frac{x}{x ^{2} - 3 x - 4} d x

\frac{1}{5} ln ∣ x + 1 ∣ + \frac{4}{5} ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{x}{x ^{2} - 3 x - 4} d x

Take the partial fraction of

\frac{x}{x ^{2} - 3 x - 4} : \frac{1}{5 ( x + 1 )} + \frac{4}{5 ( x - 4 )}

= \int \frac{1}{5 ( x + 1 )} + \frac{4}{5 ( x - 4 )} 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}{5 ( x + 1 )} d x + \int \frac{4}{5 ( x - 4 )} d x

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

\int \frac{4}{5 ( x - 4 )} d x = \frac{4}{5} ln ∣ x - 4 ∣

= \frac{1}{5} ln ∣ x + 1 ∣ + \frac{4}{5} ln ∣ x - 4 ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ x + 1 ∣ + \frac{4}{5} ln ∣ x - 4 ∣ + C

y^{^{'}} = - y (x + 1)

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

\int_{7}^{8} \frac{x}{x ^{2} + 4 x + 20} d x

\int \frac{5}{35 sin ( x ) - 12 cos ( x )} d x

\frac{d}{d x} (sinh (7 x))

What is the integral of x/(x^2-3x-4) ?
The integral of x/(x^2-3x-4) is 1/5 ln|x+1|+4/5 ln|x-4|+C