Question

$\int \frac{x^{1/3}}{(1+x^{4/3}{)}^2}dx=$

• A. $\frac{-3}{4(1+x^{4/3})}+c$
• B. $\frac{3}{4(1+x^{4/3})}+c$
• C. $\frac{1}{(1+x^{4/3})}+c$
• D. $\frac{-1}{(1+x^{4/3})}+c$

Answer 1

The correct option is A $\frac{-3}{4(1+x^{4/3})}+c$

Take $x^{\frac{2}{3}}=tan\theta$

Then, Integral, $I$ becomes $\int \frac{3}{2}\frac{tan\theta }{{sec}^2\theta }d\theta$

Now take $sec\theta =t,I=\int \frac{3}{2}\frac{dt}{t^3}$

$I=\frac{-3}{4t^2}$

Now substitute the value of $t$ and $\theta$