Question

Solve:

$\int x^{2/3}{(1+x^{5/3})}^{\frac{-13}{3}}dx$

• A. $-\frac{9}{25(1+x^{5/3}{)}^{\frac{10}{3}}}+C$
• B. $-\frac{9}{15(1+x^{5/3}{)}^{\frac{10}{3}}}+C$
• C. $-\frac{9}{50(1+x^{5/3}{)}^{\frac{10}{3}}}+C$
• D. None of these

Answer 1

The correct option is C

$-\frac{9}{50(1+x^{5/3}{)}^{\frac{10}{3}}}+C$

We have,

$I=\int x^{2/3}{(1+x^{5/3})}^{\frac{-13}{3}}dx$

Let

$t=1+x^{5/3}$

$\frac{dt}{dx}=0+\frac{5}{3}{x}^{2/3}$

$\frac{3dt}{5}=x^{2/3}dx$

Therefore,

$I=\frac{3}{5}\int t^{\frac{-13}{3}}dt$

$I=\frac{3}{5}\frac{t^{\frac{-10}{3}}}{\frac{-10}{3}}+C$

$I=-\frac{9}{50}(1+x^{5/3}{)}^{\frac{-10}{3}}+C$

$I=-\frac{9}{50(1+x^{5/3}{)}^{\frac{10}{3}}}+C$

Hence, this is the answer.