What is the integral of (25x^2)/((x-3)(x+2)^2) ?

The integral of (25x^2)/((x-3)(x+2)^2) is 16ln|x+2|+(20)/(x+2)+9ln|x-3|+C

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integral of (25x^2)/((x-3)(x+2)^2)

\int \frac{25 x ^{2}}{( x - 3 ) ( x + 2 ) ^{2}} d x

16 ln ∣ x + 2 ∣ + \frac{20}{x + 2} + 9 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{25 x ^{2}}{( x - 3 ) ( x + 2 ) ^{2}} d x

Take the constant out:

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

= 25 \cdot \int \frac{x ^{2}}{( x - 3 ) ( x + 2 ) ^{2}} d x

Apply u-substitution

: \int \frac{( u - 2 ) ^{2}}{u ^{2} ( u - 5 )} d u

= 25 \cdot \int \frac{( u - 2 ) ^{2}}{u ^{2} ( u - 5 )} d u

Take the partial fraction of

\frac{( u - 2 ) ^{2}}{u ^{2} ( u - 5 )} : \frac{16}{25 u} - \frac{4}{5 u ^{2}} + \frac{9}{25 ( u - 5 )}

= 25 \cdot \int \frac{16}{25 u} - \frac{4}{5 u ^{2}} + \frac{9}{25 ( u - 5 )} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 25 (\int \frac{16}{25 u} d u - \int \frac{4}{5 u ^{2}} d u + \int \frac{9}{25 ( u - 5 )} d u)

\int \frac{16}{25 u} d u = \frac{16}{25} ln ∣ u ∣

\int \frac{4}{5 u ^{2}} d u = - \frac{4}{5 u}

\int \frac{9}{25 ( u - 5 )} d u = \frac{9}{25} ln ∣ u - 5 ∣

= 25 (\frac{16}{25} ln ∣ u ∣ - (- \frac{4}{5 u}) + \frac{9}{25} ln ∣ u - 5 ∣)

Substitute back

u = x + 2

= 25 (\frac{16}{25} ln ∣ x + 2 ∣ - (- \frac{4}{5 ( x + 2 )}) + \frac{9}{25} ln ∣ x + 2 - 5 ∣)

Simplify

25 (\frac{16}{25} ln ∣ x + 2 ∣ - (- \frac{4}{5 ( x + 2 )}) + \frac{9}{25} ln ∣ x + 2 - 5 ∣) : 16 ln ∣ x + 2 ∣ + \frac{20}{x + 2} + 9 ln ∣ x - 3 ∣

= 16 ln ∣ x + 2 ∣ + \frac{20}{x + 2} + 9 ln ∣ x - 3 ∣

Add a constant to the solution

= 16 ln ∣ x + 2 ∣ + \frac{20}{x + 2} + 9 ln ∣ x - 3 ∣ + C

\int x arctan (x) d x

x \to 1 + lim (ln (ln (x)))

\frac{d}{d x} (e^{- 10 x})

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

x \to - \infty lim (5^{x})

What is the integral of (25x^2)/((x-3)(x+2)^2) ?
The integral of (25x^2)/((x-3)(x+2)^2) is 16ln|x+2|+(20)/(x+2)+9ln|x-3|+C