Question

$\int x^{2 / 3}\left(1 + x^{1 / 2}\right)^{\frac{- 13}{3}}dx$

• A. $\frac{3}{5}\left(1 + x^{- 1 / 2}\right)^{- 10 / 3}+C$
• B. $\frac{3}{5}\left(1 + x^{1 / 2}\right)^{- \frac{10}{3}}+C$
• C. $\frac{\left(1 + x^{1 / 2}\right)}{13}+C$
• D. None of the above

Answer 1

Answer: (A)

We have given that $\int x^{2 / 3}\left(1 + x^{1 / 2}\right)^{- 13 / 3}dx$
$=\int x^{2 / 3}\cdot x^{- 13 / 6}\left(x^{- 1 / 2} + 1\right)^{- 13 / 3}dx$
$=\int x^{- 3 / 2}\left(x^{- 1 / 2} + 1\right)^{- 13 / 3}dx$
Let $x^{- 1 / 2}+1=t$
$-\frac{1}{2}x^{- 3 / 2}dx=dt$
$x^{- 3 / 2}dt=-2dt=-2\int t^{- 13 / 3}dt$
$=-2\frac{t^{- 10 / 3}}{\left(- \frac{10}{3}\right)}=\frac{3}{5}\left(x^{- 1 / 2} + 1\right)^{- 10 / 3}+C$