What is the integral of (x+1)sqrt(x+6) ?

The integral of (x+1)sqrt(x+6) is 2/5 (x+6)^{5/2}-10/3 (x+6)^{3/2}+C

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integral of (x+1)sqrt(x+6)

\int (x + 1) x + 6 d x

\frac{2}{5} (x + 6)^{\frac{5}{2}} - \frac{10}{3} (x + 6)^{\frac{3}{2}} + C

Solution steps

\int (x + 1) x + 6 d x

Apply u-substitution

: \int u (u - 5) d u

= \int u (u - 5) d u

Expand

u (u - 5) : u^{\frac{3}{2}} - 5 u

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

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

\int 5 u d u = \frac{10}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} - \frac{10}{3} u^{\frac{3}{2}}

Substitute back

u = x + 6

= \frac{2}{5} (x + 6)^{\frac{5}{2}} - \frac{10}{3} (x + 6)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (x + 6)^{\frac{5}{2}} - \frac{10}{3} (x + 6)^{\frac{3}{2}} + C

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What is the integral of (x+1)sqrt(x+6) ?
The integral of (x+1)sqrt(x+6) is 2/5 (x+6)^{5/2}-10/3 (x+6)^{3/2}+C