What is the integral of 4xsqrt(x-6) ?

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

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integral of 4xsqrt(x-6)

\int 4 x x - 6 d x

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

Solution steps

\int 4 x x - 6 d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int x x - 6 d x

Apply u-substitution

: \int (u + 6) u d u

= 4 \cdot \int (u + 6) u d u

Expand

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

= 4 \cdot \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

= 4 (\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}}

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

Substitute back

u = x - 6

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

Add a constant to the solution

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

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