What is the integral of (x-3)/((x+4)(x+5)) ?

The integral of (x-3)/((x+4)(x+5)) is -7ln|x+4|+8ln|x+5|+C

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integral of (x-3)/((x+4)(x+5))

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

- 7 ln ∣ x + 4 ∣ + 8 ln ∣ x + 5 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x - 3}{( x + 4 ) ( x + 5 )} : - \frac{7}{x + 4} + \frac{8}{x + 5}

= \int - \frac{7}{x + 4} + \frac{8}{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

= - \int \frac{7}{x + 4} d x + \int \frac{8}{x + 5} d x

\int \frac{7}{x + 4} d x = 7 ln ∣ x + 4 ∣

\int \frac{8}{x + 5} d x = 8 ln ∣ x + 5 ∣

= - 7 ln ∣ x + 4 ∣ + 8 ln ∣ x + 5 ∣

Add a constant to the solution

= - 7 ln ∣ x + 4 ∣ + 8 ln ∣ x + 5 ∣ + C

csc (θ)

\int_{π}^{2 π} x sin (\frac{x}{2}) d x

y^{^{''}} + 12 y^{^{'}} + 36 y = t^{- 2} e^{- 6 t}

y^{^{''}} - 2 y^{^{'}} + 2 y = 0

y = 4 - x^{2}, (- 1, 3)

What is the integral of (x-3)/((x+4)(x+5)) ?
The integral of (x-3)/((x+4)(x+5)) is -7ln|x+4|+8ln|x+5|+C