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 (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

y (θ) = \frac{π}{4} sin (θ)

\frac{1 0 ^{x}}{ln ( 10 )}

\int a x^{n} d x

t^{2} \frac{d y}{d t} - t = 1 + y + t y

x \to - \infty lim (\frac{x ^{3} + x ^{2} - 3 x + 2}{x ^{2} - 3 x + 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