What is the integral of (32x^2)/((x-21)(x+7)^2) ?

The integral of (32x^2)/((x-21)(x+7)^2) is 32(9/16 ln|x-21|+7/16 ln|x+7|+7/(4(x+7)))+C

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integral of (32x^2)/((x-21)(x+7)^2)

\int \frac{32 x ^{2}}{( x - 21 ) ( x + 7 ) ^{2}} d x

32 (\frac{9}{16} ln ∣ x - 21 ∣ + \frac{7}{16} ln ∣ x + 7 ∣ + \frac{7}{4 ( x + 7 )}) + C

Solution steps

\int \frac{32 x ^{2}}{( x - 21 ) ( x + 7 ) ^{2}} d x

Take the constant out:

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

= 32 \cdot \int \frac{x ^{2}}{( x - 21 ) ( x + 7 ) ^{2}} d x

Take the partial fraction of

\frac{x ^{2}}{( x - 21 ) ( x + 7 ) ^{2}} : \frac{9}{16 ( x - 21 )} + \frac{7}{16 ( x + 7 )} - \frac{7}{4 ( x + 7 ) ^{2}}

= 32 \cdot \int \frac{9}{16 ( x - 21 )} + \frac{7}{16 ( x + 7 )} - \frac{7}{4 ( x + 7 ) ^{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

= 32 (\int \frac{9}{16 ( x - 21 )} d x + \int \frac{7}{16 ( x + 7 )} d x - \int \frac{7}{4 ( x + 7 ) ^{2}} d x)

\int \frac{9}{16 ( x - 21 )} d x = \frac{9}{16} ln ∣ x - 21 ∣

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

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

= 32 (\frac{9}{16} ln ∣ x - 21 ∣ + \frac{7}{16} ln ∣ x + 7 ∣ - (- \frac{7}{4 ( x + 7 )}))

Simplify

= 32 (\frac{9}{16} ln ∣ x - 21 ∣ + \frac{7}{16} ln ∣ x + 7 ∣ + \frac{7}{4 ( x + 7 )})

Add a constant to the solution

= 32 (\frac{9}{16} ln ∣ x - 21 ∣ + \frac{7}{16} ln ∣ x + 7 ∣ + \frac{7}{4 ( x + 7 )}) + C

2 - \frac{3}{x}, at x = 1

\frac{\partial}{\partial x} (ln (x + 4 y + 2 z))

\int_{- 2}^{3} (x^{2} - 3) d x

\frac{d}{d x} (sin (x + 2 y))

What is the integral of (32x^2)/((x-21)(x+7)^2) ?
The integral of (32x^2)/((x-21)(x+7)^2) is 32(9/16 ln|x-21|+7/16 ln|x+7|+7/(4(x+7)))+C