What is the integral of xsqrt(x-6) ?

The integral of xsqrt(x-6) is 2/5 (x-6)^{5/2}+4(x-6)^{3/2}+C

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integral of xsqrt(x-6)

\int x x - 6 d x

\frac{2}{5} (x - 6)^{\frac{5}{2}} + 4 (x - 6)^{\frac{3}{2}} + C

Solution steps

\int x x - 6 d x

Apply u-substitution

: \int (u + 6) u d u

= \int (u + 6) u d u

Expand

(u + 6) u : u^{\frac{3}{2}} + 6 u

= \int u^{\frac{3}{2}} + 6 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 6 u d u

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

\int 6 u d u = 4 u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} + 4 u^{\frac{3}{2}}

Substitute back

u = x - 6

= \frac{2}{5} (x - 6)^{\frac{5}{2}} + 4 (x - 6)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (x - 6)^{\frac{5}{2}} + 4 (x - 6)^{\frac{3}{2}} + C

\frac{d}{d x} (x sinh (x) - 8 cosh (x))

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

\int \frac{5}{3 x ^{2} - 2} d x

x \to \infty lim (- 5^{x})

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What is the integral of xsqrt(x-6) ?
The integral of xsqrt(x-6) is 2/5 (x-6)^{5/2}+4(x-6)^{3/2}+C