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: Simplify
\(\frac{x}{x^{5/2}+4x^{3/2}}=\frac{x}{x^{3/2}(x+4)}=\frac{1}{\sqrt{x}(x+4)}\).

Step 2: Substitute
Let \(t=\sqrt{x}\), so \(dx=2t\,dt\).

Step 3: Integrate
\(\int \frac{2t\,dt}{t(t^{2}+4)}=2\int \frac{dt}{t^{2}+4}=2\cdot \frac{1}{2}\tan ^{-1}\frac{t}{2}\).

Step 4: Substitute back
The result is \(\tan ^{-1}\left(\frac{\sqrt{x}}{2}\right)+c\).

Final Answer:
The integral is \(\tan ^{-1}\left(\frac{\sqrt{x}}{2}\right)+c\). \[ \boxed{\text{(D) }\tan ^{-1}\left(\dfrac{\sqrt{x}}{2}\right)+c} \]