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

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

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integral of (sqrt(x-1))^3

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

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

Solution steps

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

Simplify

(x - 1)^{3} : (x - 1)^{2} x - 1

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

(x - 1)^{2} = x - 1

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

Apply u-substitution

: \int u u d u

= \int u u d u

Simplify

u u : u^{\frac{3}{2}}

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

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

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

Substitute back

u = x - 1

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

Add a constant to the solution

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

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