int_0^3 frac{dx}{(x+2) sqrt{x+1}} = | vedclass

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(B) Let $I = \int_0^3 \frac{dx}{(x+2) \sqrt{x+1}}$.Substitute $t = \sqrt{x+1}$,so $t^2 = x+1$ and $x = t^2 - 1$.Then $dx = 2t \, dt$.When $x = 0$,$t =

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$2 	an^{-1}\left(\frac{1}{3}ight)$

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(B) Let $I = \int_0^3 \frac{dx}{(x+2) \sqrt{x+1}}$.Substitute $t = \sqrt{x+1}$,so $t^2 = x+1$ and $x = t^2 - 1$.Then $dx = 2t \, dt$.When $x = 0$,$t = 1$. When $x = 3$,$t = 2$.The integral becomes $I = \int_1^2 \frac{2t \, dt}{(t^2 - 1 + 2) t} = \int_1^2 \frac{2 \, dt}{t^2 + 1}$.Integrating,we get $I = [2 	an^{-1}(t)]_1^2 = 2(	an^{-1}(2) - 	an^{-1}(1))$.Using the formula $	an^{-1}(A) - 	an^{-1}(B) = 	an^{-1}\left(\frac{A-B}{1+AB}ight)$,$I = 2 	an^{-1}\left(\frac{2-1}{1+2(1)}ight) = 2 	an^{-1}\left(\frac{1}{3}ight)$.

$\int_0^3 \frac{dx}{(x+2) \sqrt{x+1}} = $

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