Question

$\int \frac{dx}{\sqrt{x+1} + \sqrt{x}} =$

• A. $\frac{2}{3}[(x+1)^{\frac{3}{2}} - (x)^{\frac{3}{2}}] + c$
• B. $[(x+1) + (x)] + c$
• C. $\frac{3}{2}[(x+1)^{\frac{3}{2}} - (x)^{\frac{3}{2}}] + c$
• D. $\frac{3}{2}[(x+1)^{\frac{3}{2}} + (x)^{\frac{3}{2}}] + c$

Hint:

Rationalization isn't just for limits; it's a powerful tool for simplifying radical integrals too.

Answer 1

The Correct Option is A

Step 1: Concept
Rationalize the denominator by multiplying the numerator and denominator by the conjugate $(\sqrt{x+1} - \sqrt{x})$.

Step 2: Meaning
The denominator becomes $(x+1) - x = 1$, simplifying the integral significantly.

Step 3: Analysis
The integral becomes $\int (\sqrt{x+1} - \sqrt{x}) dx = \int ((x+1)^{1/2} - x^{1/2}) dx$.

Step 4: Conclusion
Integrating gives $\frac{(x+1)^{3/2}}{3/2} - \frac{x^{3/2}}{3/2} = \frac{2}{3}[(x+1)^{3/2} - x^{3/2}] + c$.
Final Answer: (A)