What is the integral of (37-11x)/((x+1)(x-2)(x-3)) ?

The integral of (37-11x)/((x+1)(x-2)(x-3)) is 4ln|x+1|-5ln|x-2|+ln|x-3|+C

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integral of (37-11x)/((x+1)(x-2)(x-3))

\int \frac{37 - 11 x}{( x + 1 ) ( x - 2 ) ( x - 3 )} d x

4 ln ∣ x + 1 ∣ - 5 ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{37 - 11 x}{( x + 1 ) ( x - 2 ) ( x - 3 )} d x

Take the partial fraction of

\frac{37 - 11 x}{( x + 1 ) ( x - 2 ) ( x - 3 )} : \frac{4}{x + 1} - \frac{5}{x - 2} + \frac{1}{x - 3}

= \int \frac{4}{x + 1} - \frac{5}{x - 2} + \frac{1}{x - 3} 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{4}{x + 1} d x - \int \frac{5}{x - 2} d x + \int \frac{1}{x - 3} d x

\int \frac{4}{x + 1} d x = 4 ln ∣ x + 1 ∣

\int \frac{5}{x - 2} d x = 5 ln ∣ x - 2 ∣

\int \frac{1}{x - 3} d x = ln ∣ x - 3 ∣

= 4 ln ∣ x + 1 ∣ - 5 ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣

Add a constant to the solution

= 4 ln ∣ x + 1 ∣ - 5 ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ + C

y^{^{''}} + 3 y^{^{'}} = 4 x - 5

y^{^{''}} + 4 y^{^{'}} + 4 = 0, y^{^{'}} (0) = 1

x \to \infty lim (- \frac{1}{x})

f (x) = cos (x), - \frac{π}{2}, \frac{π}{2}

x y^{^{'}} = y + y^{3}

What is the integral of (37-11x)/((x+1)(x-2)(x-3)) ?
The integral of (37-11x)/((x+1)(x-2)(x-3)) is 4ln|x+1|-5ln|x-2|+ln|x-3|+C