Find ∫ x/(√(x + 1)) dx for x > -1. Write the answer as a(x + 1)^(3/2) + b(x + 1)^(1/2) + C…

Step 1. Substitute. Let $u = x + 1$, so $x = u - 1$ and $dx = du$. The integral is $\displaystyle\int \dfrac{u - 1}{\sqrt{u}}\,du = \int \left(u^{1/2} - u^{-1/2}\right)du$.

Step 2. Integrate. The result is $\tfrac{2}{3}u^{3/2} - 2u^{1/2} + C$. So $a = \tfrac{2}{3} = 0.6667$ and $b = -2$.

Step 3. Use $F(0) = 0$. At $x = 0$, $u = 1$, so $\tfrac{2}{3} - 2 + C = 0$ and $C = \tfrac{4}{3}$.

Step 4. Evaluate. At $x = 3$, $u = 4$, so $F(3) = \tfrac{2}{3}(8) - 2(2) + \tfrac{4}{3} = \tfrac{8}{3} = 2.6667$.