What is the integral of (2x+5)/((x-4)(x+1)) ?

The integral of (2x+5)/((x-4)(x+1)) is 13/5 ln|x-4|-3/5 ln|x+1|+C

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integral of (2x+5)/((x-4)(x+1))

\int \frac{2 x + 5}{( x - 4 ) ( x + 1 )} d x

\frac{13}{5} ln ∣ x - 4 ∣ - \frac{3}{5} ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{2 x + 5}{( x - 4 ) ( x + 1 )} d x

Take the partial fraction of

\frac{2 x + 5}{( x - 4 ) ( x + 1 )} : \frac{13}{5 ( x - 4 )} - \frac{3}{5 ( x + 1 )}

= \int \frac{13}{5 ( x - 4 )} - \frac{3}{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{13}{5 ( x - 4 )} d x - \int \frac{3}{5 ( x + 1 )} d x

\int \frac{13}{5 ( x - 4 )} d x = \frac{13}{5} ln ∣ x - 4 ∣

\int \frac{3}{5 ( x + 1 )} d x = \frac{3}{5} ln ∣ x + 1 ∣

= \frac{13}{5} ln ∣ x - 4 ∣ - \frac{3}{5} ln ∣ x + 1 ∣

Add a constant to the solution

= \frac{13}{5} ln ∣ x - 4 ∣ - \frac{3}{5} ln ∣ x + 1 ∣ + C

y = 3 x^{3}

y^{^{''}} + 3 y^{^{'}} + 2 y = sin (t)

\int (3 - 2 x) sin (5 x) d x

\int cot (9 x) d x

y = x^{5} \cdot e^{x}

What is the integral of (2x+5)/((x-4)(x+1)) ?
The integral of (2x+5)/((x-4)(x+1)) is 13/5 ln|x-4|-3/5 ln|x+1|+C