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

The integral of (x^2-3x+2)/(sqrt(x-1)) is 2/5 x(x-1)^{3/2}-16/15 (x-1)^{3/2}+C

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

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

\frac{2}{5} x (x - 1)^{\frac{3}{2}} - \frac{16}{15} (x - 1)^{\frac{3}{2}} + C

Solution steps

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

Simplify

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

= \int (x - 2) x - 1 d x

Apply Integration By Parts

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

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

\int \frac{2}{3} (x - 1)^{\frac{3}{2}} d x = \frac{2}{15} (2 x (x - 1)^{\frac{3}{2}} - 2 (x - 1)^{\frac{3}{2}})

= \frac{2}{3} (x - 2) (x - 1)^{\frac{3}{2}} - \frac{2}{15} (2 x (x - 1)^{\frac{3}{2}} - 2 (x - 1)^{\frac{3}{2}})

Simplify

\frac{2}{3} (x - 2) (x - 1)^{\frac{3}{2}} - \frac{2}{15} (2 x (x - 1)^{\frac{3}{2}} - 2 (x - 1)^{\frac{3}{2}}) : \frac{2}{5} x (x - 1)^{\frac{3}{2}} - \frac{16}{15} (x - 1)^{\frac{3}{2}}

= \frac{2}{5} x (x - 1)^{\frac{3}{2}} - \frac{16}{15} (x - 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} x (x - 1)^{\frac{3}{2}} - \frac{16}{15} (x - 1)^{\frac{3}{2}} + C

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What is the integral of (x^2-3x+2)/(sqrt(x-1)) ?
The integral of (x^2-3x+2)/(sqrt(x-1)) is 2/5 x(x-1)^{3/2}-16/15 (x-1)^{3/2}+C