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

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

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