Question

\(\int \frac{x}{x^{5/2}+4x^{3/2}}\ dx=\ldots\)

• A. \(3\,\log |x^{5/2}+4x^{3/2}|+c\)
• B. \(\frac{1}{2}\log |x+4|+c\)
• C. \(4\,\sec (x^{3/2})+\log \left(\tan \left(\frac{x}{2}\right)\right)+c\)
• D. \(\tan ^{-1}\left(\frac{\sqrt{x}}{2}\right)+c\)

Hint:

Cancel the common power of x to get a form with the square root of x, then substitute t = sqrt(x).

Answer 1

The Correct Option is D

Step 1: Differentiate the answer
$\frac{d}{dx}\tan ^{-1}\frac{\sqrt{x}}{2}=\frac{1}{1+x/4}\cdot \frac{1}{4\sqrt{x}}$.

Step 2: Simplify
$=\frac{4}{4+x}\cdot \frac{1}{4\sqrt{x}}=\frac{1}{\sqrt{x}(x+4)}$.

Step 3: Compare
This equals the simplified integrand, so the option is correct.

Final Answer:
Differentiating $\tan ^{-1}\frac{\sqrt{x}}{2}$ returns the integrand. \[ \boxed{\text{(D) }\tan ^{-1}\left(\dfrac{\sqrt{x}}{2}\right)+c} \]