What is the integral of (16x^2)/((x-15)(x+5)^2) ?

The integral of (16x^2)/((x-15)(x+5)^2) is 9ln|x-15|+7ln|x+5|+(20)/(x+5)+C

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integral of (16x^2)/((x-15)(x+5)^2)

\int \frac{16 x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} d x

9 ln ∣ x - 15 ∣ + 7 ln ∣ x + 5 ∣ + \frac{20}{x + 5} + C

Solution steps

\int \frac{16 x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} d x

Take the constant out:

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

= 16 \cdot \int \frac{x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} d x

Take the partial fraction of

\frac{x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} : \frac{9}{16 ( x - 15 )} + \frac{7}{16 ( x + 5 )} - \frac{5}{4 ( x + 5 ) ^{2}}

= 16 \cdot \int \frac{9}{16 ( x - 15 )} + \frac{7}{16 ( x + 5 )} - \frac{5}{4 ( x + 5 ) ^{2}} 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

= 16 (\int \frac{9}{16 ( x - 15 )} d x + \int \frac{7}{16 ( x + 5 )} d x - \int \frac{5}{4 ( x + 5 ) ^{2}} d x)

\int \frac{9}{16 ( x - 15 )} d x = \frac{9}{16} ln ∣ x - 15 ∣

\int \frac{7}{16 ( x + 5 )} d x = \frac{7}{16} ln ∣ x + 5 ∣

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

= 16 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ - (- \frac{5}{4 ( x + 5 )}))

Simplify

16 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ - (- \frac{5}{4 ( x + 5 )})) : 9 ln ∣ x - 15 ∣ + 7 ln ∣ x + 5 ∣ + \frac{20}{x + 5}

= 9 ln ∣ x - 15 ∣ + 7 ln ∣ x + 5 ∣ + \frac{20}{x + 5}

Add a constant to the solution

= 9 ln ∣ x - 15 ∣ + 7 ln ∣ x + 5 ∣ + \frac{20}{x + 5} + C

\int \frac{x ^{3}}{3 + x ^{2}} d x

\frac{d}{d x} (3^{4 x})

f (x) = x^{2} - 3, (- 4, 13)

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

\int_{0}^{1} x e^{3 x^{2}} d x

What is the integral of (16x^2)/((x-15)(x+5)^2) ?
The integral of (16x^2)/((x-15)(x+5)^2) is 9ln|x-15|+7ln|x+5|+(20)/(x+5)+C