Integrals Involving Roots — Question 4

Question 4

Rationalize the integrand and evaluate $$\int\frac{dx}{\sqrt{x+1}-\sqrt{x}}.$$

Question 4 – Solution

Step 1: Multiply by the conjugate. $$\begin{align*} \frac1{\sqrt{x+1}-\sqrt{x}} &=\frac1{\sqrt{x+1}-\sqrt{x}} \cdot\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}\\ &=\frac{\sqrt{x+1}+\sqrt{x}}{(x+1)-x}\\ &=\sqrt{x+1}+\sqrt{x}. \end{align*}$$ Step 2: Integrate both terms. $$\begin{align*} I&=\int (x+1)^{1/2}\,dx+\int x^{1/2}\,dx\\ &=\frac{(x+1)^{3/2}}{3/2}+\frac{x^{3/2}}{3/2}+C\\ &=\frac23(x+1)^{3/2}+\frac23x^{3/2}+C. \end{align*}$$ $$\boxed{\frac23(x+1)^{3/2}+\frac23x^{3/2}+C}$$