What is the integral of ysqrt(y+7) ?

The integral of ysqrt(y+7) is 2/5 (y+7)^{5/2}-14/3 (y+7)^{3/2}+C

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integral of ysqrt(y+7)

\int y y + 7 d y

\frac{2}{5} (y + 7)^{\frac{5}{2}} - \frac{14}{3} (y + 7)^{\frac{3}{2}} + C

Solution steps

\int y y + 7 d y

Apply u-substitution

: \int (u - 7) u d u

= \int (u - 7) u d u

Expand

(u - 7) u : u^{\frac{3}{2}} - 7 u

= \int u^{\frac{3}{2}} - 7 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 7 u d u

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

\int 7 u d u = \frac{14}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} - \frac{14}{3} u^{\frac{3}{2}}

Substitute back

u = y + 7

= \frac{2}{5} (y + 7)^{\frac{5}{2}} - \frac{14}{3} (y + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (y + 7)^{\frac{5}{2}} - \frac{14}{3} (y + 7)^{\frac{3}{2}} + C

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What is the integral of ysqrt(y+7) ?
The integral of ysqrt(y+7) is 2/5 (y+7)^{5/2}-14/3 (y+7)^{3/2}+C