What is the integral of ysqrt(y+8) ?

The integral of ysqrt(y+8) is 2/5 (y+8)^{5/2}-16/3 (y+8)^{3/2}+C

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integral of ysqrt(y+8)

\int y y + 8 d y

\frac{2}{5} (y + 8)^{\frac{5}{2}} - \frac{16}{3} (y + 8)^{\frac{3}{2}} + C

Solution steps

\int y y + 8 d y

Apply u-substitution

: \int (u - 8) u d u

= \int (u - 8) u d u

Expand

(u - 8) u : u^{\frac{3}{2}} - 8 u

= \int u^{\frac{3}{2}} - 8 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 8 u d u

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

\int 8 u d u = \frac{16}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} - \frac{16}{3} u^{\frac{3}{2}}

Substitute back

u = y + 8

= \frac{2}{5} (y + 8)^{\frac{5}{2}} - \frac{16}{3} (y + 8)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (y + 8)^{\frac{5}{2}} - \frac{16}{3} (y + 8)^{\frac{3}{2}} + C

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What is the integral of ysqrt(y+8) ?
The integral of ysqrt(y+8) is 2/5 (y+8)^{5/2}-16/3 (y+8)^{3/2}+C