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 $x^{-1/2}+1=t$
$-\frac{1}{2}{x}^{-3/2}dx=dt$
$x^{-3/2}dt=-2dt=-2\int t^{-13/3}dt$
$=-2\frac{t^{-10/3}}{\left(-\frac{10}{3}\right)}=\frac{3}{5}{\left(x^{-1/2}+1\right)}^{-10/3}+C$