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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\frac{d y}{d x} = \frac{2 y}{x + 1}

x \to 1 lim (x + 4)

n = 1 \sum \infty \frac{n}{( 5 ^{n} )}

\int x^{2} \cdot sin (2 x) d x

\frac{d}{d x} ((4 x^{2} + 1)^{- 50})

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