What is the integral of rsqrt(r+3) ?

The integral of rsqrt(r+3) is 2/5 (r+3)^{5/2}-2(r+3)^{3/2}+C

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integral of rsqrt(r+3)

\int r r + 3 d r

\frac{2}{5} (r + 3)^{\frac{5}{2}} - 2 (r + 3)^{\frac{3}{2}} + C

Solution steps

\int r r + 3 d r

Apply u-substitution

: \int (u - 3) u d u

= \int (u - 3) u d u

Expand

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

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

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

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

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

Substitute back

u = r + 3

= \frac{2}{5} (r + 3)^{\frac{5}{2}} - 2 (r + 3)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (r + 3)^{\frac{5}{2}} - 2 (r + 3)^{\frac{3}{2}} + C

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What is the integral of rsqrt(r+3) ?
The integral of rsqrt(r+3) is 2/5 (r+3)^{5/2}-2(r+3)^{3/2}+C