What is the integral of (4x^2+3)/((x-5)^2) ?

The integral of (4x^2+3)/((x-5)^2) is 4(-(x^2)/(x-5)+2(x-5+5ln|x-5|))-3/(x-5)+C

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integral of (4x^2+3)/((x-5)^2)

\int \frac{4 x ^{2} + 3}{( x - 5 ) ^{2}} d x

4 (- \frac{x ^{2}}{x - 5} + 2 (x - 5 + 5 ln ∣ x - 5 ∣)) - \frac{3}{x - 5} + C

Solution steps

\int \frac{4 x ^{2} + 3}{( x - 5 ) ^{2}} d x

Expand

\frac{4 x ^{2} + 3}{( x - 5 ) ^{2}} : \frac{4 x ^{2}}{( x - 5 ) ^{2}} + \frac{3}{( x - 5 ) ^{2}}

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{4 x ^{2}}{( x - 5 ) ^{2}} d x + \int \frac{3}{( x - 5 ) ^{2}} d x

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

\int \frac{3}{( x - 5 ) ^{2}} d x = - \frac{3}{x - 5}

= 4 (- \frac{x ^{2}}{x - 5} + 2 (x - 5 + 5 ln ∣ x - 5 ∣)) - \frac{3}{x - 5}

Add a constant to the solution

= 4 (- \frac{x ^{2}}{x - 5} + 2 (x - 5 + 5 ln ∣ x - 5 ∣)) - \frac{3}{x - 5} + C

\frac{d}{d x} (\frac{( x + 5 ) ( x ^{2} - 5 x + 25 )}{x ^{3}})

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

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

f (x) = 2000 (x^{2} + 5)^{\frac{1}{3}}

\int_{0}^{2} e^{- x^{2}} d x

What is the integral of (4x^2+3)/((x-5)^2) ?
The integral of (4x^2+3)/((x-5)^2) is 4(-(x^2)/(x-5)+2(x-5+5ln|x-5|))-3/(x-5)+C