What is the integral of xsqrt(x-18) ?

The integral of xsqrt(x-18) is 2/5 (x-18)^{5/2}+12(x-18)^{3/2}+C

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integral of xsqrt(x-18)

\int x x - 18 d x

\frac{2}{5} (x - 18)^{\frac{5}{2}} + 12 (x - 18)^{\frac{3}{2}} + C

Solution steps

\int x x - 18 d x

Apply u-substitution

: \int (u + 18) u d u

= \int (u + 18) u d u

Expand

(u + 18) u : u^{\frac{3}{2}} + 18 u

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

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

\int 18 u d u = 12 u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} + 12 u^{\frac{3}{2}}

Substitute back

u = x - 18

= \frac{2}{5} (x - 18)^{\frac{5}{2}} + 12 (x - 18)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (x - 18)^{\frac{5}{2}} + 12 (x - 18)^{\frac{3}{2}} + C

\frac{d}{d x} (4 e^{- 2 x^{2}})

(- 2.2) (2.3)

\frac{cos ( x )}{x ^{3}}

\int_{0}^{1} y 3 y + 1 d y

\frac{d}{d x} ((5 - 3 x)^{2})

What is the integral of xsqrt(x-18) ?
The integral of xsqrt(x-18) is 2/5 (x-18)^{5/2}+12(x-18)^{3/2}+C