Question

Evaluate $\int \frac{x^{5}dx}{\sqrt{(1+x^3)}}=$

• A. $\frac{2}{3}\sqrt{(1+x^3)+(x^2+2)}$
• B. $\frac{2}{9}\sqrt{(1+x^3)-(x^3-4)}$
• C. $\frac{2}{9}\sqrt{(1+x^3)(x^3+4)}$
• D. $\frac{2}{3}\sqrt{\frac{{(1+x^3)}^3}{3}-(1+x^3)}$

Answer 1

The correct option is D $\frac{2}{3}\sqrt{\frac{{(1+x^3)}^3}{3}-(1+x^3)}$

$\int \frac{x^{5}dx}{\sqrt{(1+x^3)}}$=$x^3\cdot \int \frac{x^{2}dx}{\sqrt{(1+x^3)}}$

Let u=$\sqrt{1}+x^3$

$u^2=1+x^3$

$x^3$=$(u^2-1)$

$3x^{2}dx$=$2udu$

$\frac{2}{3}\int (u^2-1)du$

$\frac{2}{3}[\frac{u^3}{3}-\mu ]$

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

$\frac{2}{3}\sqrt{\frac{(1+x^3{)}^3}{3}-(1+x^3)}+C$(Ans)