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

The integral of (2x^2+1)/((x-3)^3) is 2ln|x-3|-(12)/(x-3)-9/((x-3)^2)-1/(2(x-3)^2)+C

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integral of (2x^2+1)/((x-3)^3)

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

2 ln ∣ x - 3 ∣ - \frac{12}{x - 3} - \frac{9}{( x - 3 ) ^{2}} - \frac{1}{2 ( x - 3 ) ^{2}} + C

Solution steps

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

Expand

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

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

\int \frac{2 x ^{2}}{( x - 3 ) ^{3}} d x = 2 ln ∣ x - 3 ∣ - \frac{12}{x - 3} - \frac{9}{( x - 3 ) ^{2}}

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

= 2 ln ∣ x - 3 ∣ - \frac{12}{x - 3} - \frac{9}{( x - 3 ) ^{2}} - \frac{1}{2 ( x - 3 ) ^{2}}

Add a constant to the solution

= 2 ln ∣ x - 3 ∣ - \frac{12}{x - 3} - \frac{9}{( x - 3 ) ^{2}} - \frac{1}{2 ( x - 3 ) ^{2}} + C

\int x (x + 5)^{2} d x

x y^{^{'}} = 1 - y^{2}

f (x) = - 2 ln (x)

x \to \infty lim (9 + \frac{4}{x})

\frac{d}{d x} (- 512 sin (2 x))

What is the integral of (2x^2+1)/((x-3)^3) ?
The integral of (2x^2+1)/((x-3)^3) is 2ln|x-3|-(12)/(x-3)-9/((x-3)^2)-1/(2(x-3)^2)+C