What is the integral of (2x)/(x^2-3x-10) ?

The integral of (2x)/(x^2-3x-10) is 2(2/7 ln|x+2|+5/7 ln|x-5|)+C

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integral of (2x)/(x^2-3x-10)

\int \frac{2 x}{x ^{2} - 3 x - 10} d x

2 (\frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x - 5 ∣) + C

Solution steps

\int \frac{2 x}{x ^{2} - 3 x - 10} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{x ^{2} - 3 x - 10} d x

Take the partial fraction of

\frac{x}{x ^{2} - 3 x - 10} : \frac{2}{7 ( x + 2 )} + \frac{5}{7 ( x - 5 )}

= 2 \cdot \int \frac{2}{7 ( x + 2 )} + \frac{5}{7 ( x - 5 )} 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

= 2 (\int \frac{2}{7 ( x + 2 )} d x + \int \frac{5}{7 ( x - 5 )} d x)

\int \frac{2}{7 ( x + 2 )} d x = \frac{2}{7} ln ∣ x + 2 ∣

\int \frac{5}{7 ( x - 5 )} d x = \frac{5}{7} ln ∣ x - 5 ∣

= 2 (\frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x - 5 ∣)

Add a constant to the solution

= 2 (\frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x - 5 ∣) + C

\frac{\partial}{\partial y} (x^{\frac{1}{2}} \cdot y^{\frac{1}{2}})

\frac{d}{d t} (\frac{cos ( t )}{sin ( t )})

\frac{d}{d x} (x arctan (2 x))

x \to 0 - lim (x csc (x))

\frac{d ^{2}}{d x ^{2}} (cos (- 2 x))

What is the integral of (2x)/(x^2-3x-10) ?
The integral of (2x)/(x^2-3x-10) is 2(2/7 ln|x+2|+5/7 ln|x-5|)+C