The value of integral \( \int\displaystyle\frac{\sqrt{x}}{\sqrt{1-x^{3/2}}}\,dx \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( -\displaystyle\frac{4}{3}\sqrt{1-x^{3/2}}+C \) : where \( C \) is an arbitrary constant
The derivative of \( x^{3/2} \) is \( \frac{3}{2}\sqrt{x} \), and \( \sqrt{x} \) is already in the numerator, so substitute \( t=1-x^{3/2} \).
\( \displaystyle dt=-\frac{3}{2}\sqrt{x}\,dx \)
\( \displaystyle \sqrt{x}\,dx=-\frac{2}{3}\,dt \)
\( \displaystyle \int\frac{\sqrt{x}}{\sqrt{1-x^{3/2}}}\,dx \)
\( \displaystyle =-\frac{2}{3}\int t^{-1/2}\,dt \)
\( \displaystyle =-\frac{2}{3}\cdot 2t^{1/2}+C \)
\( \displaystyle =-\frac{4}{3}\sqrt{1-x^{3/2}}+C \)
Why the other options are incorrect
Option (1) \( -\displaystyle\frac{2}{3}\sqrt{1-x^{3/2}}+C \) : where \( C \) is an arbitrary constant — Incorrect. This takes \( \int t^{-1/2}\,dt \) as \( t^{1/2} \) and loses the factor 2; in fact \( \int t^{-1/2}\,dt=\frac{t^{1/2}}{1/2}=2t^{1/2} \).
Option (2) \( \displaystyle\frac{4}{3}\sqrt{1-x^{3/2}}+C \) : where \( C \) is an arbitrary constant — Incorrect. The sign is wrong; \( dt=-\frac{3}{2}\sqrt{x}\,dx \) brings a minus sign.
Option (4) \( \displaystyle\frac{2}{3}\sqrt{1-x^{3/2}}+C \) : where \( C \) is an arbitrary constant — Incorrect. Both the minus sign and the factor 2 are missing.