What is the integral of (4x^2-21x)/((x-3)^2(2x+3)) ?

The integral of (4x^2-21x)/((x-3)^2(2x+3)) is ln|x-3|+3/(x-3)+ln|2x+3|+C

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integral of (4x^2-21x)/((x-3)^2(2x+3))

\int \frac{4 x ^{2} - 21 x}{( x - 3 ) ^{2} ( 2 x + 3 )} d x

ln ∣ x - 3 ∣ + \frac{3}{x - 3} + ln ∣ 2 x + 3 ∣ + C

Solution steps

\int \frac{4 x ^{2} - 21 x}{( x - 3 ) ^{2} ( 2 x + 3 )} d x

Take the partial fraction of

\frac{4 x ^{2} - 21 x}{( x - 3 ) ^{2} ( 2 x + 3 )} : \frac{1}{x - 3} - \frac{3}{( x - 3 ) ^{2}} + \frac{2}{2 x + 3}

= \int \frac{1}{x - 3} - \frac{3}{( x - 3 ) ^{2}} + \frac{2}{2 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{1}{x - 3} d x - \int \frac{3}{( x - 3 ) ^{2}} d x + \int \frac{2}{2 x + 3} d x

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

\int \frac{3}{( x - 3 ) ^{2}} d x = - \frac{3}{x - 3}

\int \frac{2}{2 x + 3} d x = ln ∣ 2 x + 3 ∣

= ln ∣ x - 3 ∣ - (- \frac{3}{x - 3}) + ln ∣ 2 x + 3 ∣

Simplify

= ln ∣ x - 3 ∣ + \frac{3}{x - 3} + ln ∣ 2 x + 3 ∣

Add a constant to the solution

= ln ∣ x - 3 ∣ + \frac{3}{x - 3} + ln ∣ 2 x + 3 ∣ + C

\int \frac{ln ( x ) - 1}{x ^{2}} d x

\frac{1}{y ^{4}}

\int 40

\frac{d}{d x} (tan (x + \frac{π}{4}))

\frac{s + 1}{s ^{2} + s + 3}

What is the integral of (4x^2-21x)/((x-3)^2(2x+3)) ?
The integral of (4x^2-21x)/((x-3)^2(2x+3)) is ln|x-3|+3/(x-3)+ln|2x+3|+C