\(\int\displaystyle\frac{(1 + x)^2}{\sqrt{x}}\,dx\) is equal to (where C is a constant of integration)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle 2\sqrt{x} + \frac{4}{3}x^{3/2} + \frac{2}{5}x^{5/2} + C\)
Expanding the square and dividing each term by \(\sqrt{x} = x^{1/2}\):
\( \displaystyle \frac{(1 + x)^2}{\sqrt{x}} = \frac{1 + 2x + x^2}{x^{1/2}} \)
\( = x^{-1/2} + 2x^{1/2} + x^{3/2} \)
Integrating each power with \(\int x^n\,dx = \frac{x^{n+1}}{n+1}\):
\( \displaystyle \int x^{-1/2}\,dx = 2x^{1/2} \)
\( \displaystyle \int 2x^{1/2}\,dx = \frac{4}{3}x^{3/2} \)
\( \displaystyle \int x^{3/2}\,dx = \frac{2}{5}x^{5/2} \)
So the integral is \(2\sqrt{x} + \frac{4}{3}x^{3/2} + \frac{2}{5}x^{5/2} + C\).