Question

$\int_{}^{}\frac{x^{5}}{\sqrt{1 + x^{3}}}dx$

• A. $\frac{2}{9}(1 + x^{3})^{3/2} + c$
• B. $\frac{2}{9}(1 + x^{3})^{3/2} + \frac{2}{3}(1 + x^{3})^{1/2} + c$
• C. $\frac{2}{9}(1 + x^{3})^{3/2} - \frac{2}{3}(1 + x^{3})^{1/2} + C$
• D. None of these

Answer 1

Answer: (C)

$\frac{2}{9}(1 + x^{3})^{3/2} - \frac{2}{3}(1 + x^{3})^{1/2} + C$

Answer 2

$\int_{}^{}{\frac{x^{5}}{\sqrt{1 + x^{3}}}dx} = \int_{}^{}{\frac{x^{3}.x^{2}}{\sqrt{1 + x^{3}}}dx}$. Put $1 + x^{3} = t$ $\Rightarrow 3x^{2}dx = dt$ $\Rightarrow x^{2}dx = \frac{dt}{3}$

$= \frac{1}{3}\int_{}^{}{\frac{(t - 1)}{\sqrt{t}}dt = \frac{1}{3}\int_{}^{}\left( \sqrt{t} - \frac{1}{\sqrt{t}} \right)}dt$ $= \frac{1}{3}\left\lbrack \frac{t^{3/2}}{3/2} - 2\sqrt{t} \right\rbrack + c$

$= \frac{2}{9}(1 + x^{3})^{3/2} - \frac{2}{3}(1 + x^{3})^{1/2} + c$