What is the integral of ysqrt(y+6) ?

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

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integral of ysqrt(y+6)

\int y y + 6 d y

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

Solution steps

\int y y + 6 d y

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 = y + 6

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

Add a constant to the solution

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

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