Question:
Evaluate the integral
\[
\int \frac{dx}{(2ax + x^2)^{3/2}}
\]
Evaluate the integral
\[
\int \frac{dx}{(2ax + x^2)^{3/2}}
\]
Show Hint
Use known standard integral formulas for rational powers involving linear + quadratic terms.
Updated On: May 13, 2025
- \( \frac{1}{a^2} \left( \frac{x + a}{\sqrt{2ax + x^2}} \right) + C \)
- \( \frac{1}{a^2} \left( \frac{x - a}{\sqrt{2ax + x^2}} \right) + C \)
- \( -\frac{1}{a^2} \left( \frac{x - a}{\sqrt{2ax + x^2}} \right) + C \)
- \( -\frac{1}{a^2} \left( \frac{x + a}{\sqrt{2ax + x^2}} \right) + C \)
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The Correct Option is D
Solution and Explanation
Let \( I = \int \frac{dx}{(2ax + x^2)^{3/2}} \Rightarrow \text{Use substitution: } u = \sqrt{2ax + x^2} \)
Alternatively, use standard integral:
\[
\int \frac{dx}{(x^2 + 2ax)^{3/2}} = -\frac{x + a}{a^2 \sqrt{x^2 + 2ax}} + C
\Rightarrow \boxed{-\frac{1}{a^2} \left( \frac{x + a}{\sqrt{2ax + x^2}} \right) + C}
\]
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