What is the integral of x*sqrt(x-1) ?

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

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integral of x*sqrt(x-1)

\int x \cdot x - 1 d x

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

Solution steps

\int x x - 1 d x

Apply u-substitution

: \int (u + 1) u d u

= \int (u + 1) u d u

Expand

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

= \int u^{\frac{3}{2}} + u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{\frac{3}{2}} d u + \int u d u

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = x - 1

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

Add a constant to the solution

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

x \to 0 lim (s g (s, x) n (x))

\int sin (5 x) d (5 x) d x

x \to - 5 + lim (\frac{∣ 5 - x ∣}{5 - x})

2 sin (x^{2})

\frac{2 s ^{2} + 4 s + 5}{s ( s + 1 )}

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