Question

Evaluate the integral \[ \int \frac{dx}{(2ax + x^2)^{3/2}} \]

• A. \( \frac{1}{a^2} \left( \frac{x + a}{\sqrt{2ax + x^2}} \right) + C \)
• B. \( \frac{1}{a^2} \left( \frac{x - a}{\sqrt{2ax + x^2}} \right) + C \)
• C. \( -\frac{1}{a^2} \left( \frac{x - a}{\sqrt{2ax + x^2}} \right) + C \)
• D. \( -\frac{1}{a^2} \left( \frac{x + a}{\sqrt{2ax + x^2}} \right) + C \)

Hint:

Use known standard integral formulas for rational powers involving linear + quadratic terms.

Answer 1

The Correct Option is D

Let \( I = \int \frac{dx}{(2ax + x^2)^{3/2}} \Rightarrow \text{Use substitution: } u = \sqrt{2ax + x^2} \) Alternatively, use standard integral: \[ \int \frac{dx}{(x^2 + 2ax)^{3/2}} = -\frac{x + a}{a^2 \sqrt{x^2 + 2ax}} + C \Rightarrow \boxed{-\frac{1}{a^2} \left( \frac{x + a}{\sqrt{2ax + x^2}} \right) + C} \]