What is the integral of xsqrt(x-27) ?

The integral of xsqrt(x-27) is 2/5 (x-27)^{5/2}+18(x-27)^{3/2}+C

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integral of xsqrt(x-27)

\int x x - 27 d x

\frac{2}{5} (x - 27)^{\frac{5}{2}} + 18 (x - 27)^{\frac{3}{2}} + C

Solution steps

\int x x - 27 d x

Apply u-substitution

: \int (u + 27) u d u

= \int (u + 27) u d u

Expand

(u + 27) u : u^{\frac{3}{2}} + 27 u

= \int u^{\frac{3}{2}} + 27 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 27 u d u

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

\int 27 u d u = 18 u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} + 18 u^{\frac{3}{2}}

Substitute back

u = x - 27

= \frac{2}{5} (x - 27)^{\frac{5}{2}} + 18 (x - 27)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (x - 27)^{\frac{5}{2}} + 18 (x - 27)^{\frac{3}{2}} + C

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What is the integral of xsqrt(x-27) ?
The integral of xsqrt(x-27) is 2/5 (x-27)^{5/2}+18(x-27)^{3/2}+C