Question

If $\int x^5(1+x^3{)}^{\frac{2}{3}}dx=A(1+x^3{)}^{\frac{8}{3}}+B(1+x^3{)}^{\frac{5}{3}}+C,then$

• A. $A=\frac{1}{4},B=\frac{1}{5}$
• B. $A=\frac{1}{8},B=\frac{-1}{5}$
• C. $A=\frac{-1}{8},B=\frac{1}{5}$
• D. None of these

Answer 1

The correct option is B $A=\frac{1}{8},B=\frac{-1}{5}$

Let $1+x^3=t^2$
$3x^{2}dx=2tdt.\therefore \int x^5(1+x^3{)}^{\frac{2}{3}}dx\int (t^2-1\left)\right(t^2{)}^{\frac{2}{3}}{x}^{2}dx\int (t^2-1)(t^2{)}^{\frac{2}{3}}\frac{2t}{3}dt=\frac{2}{3}\int (t^2-1)t^{\frac{7}{3}}dt=\frac{2}{3}\int \left(t^{\frac{13}{3}-t^{\frac{7}{3}}}\right)dt=\frac{2}{3}(\frac{3}{16}{t}^{\frac{16}{3}}-\frac{3}{10}{t}^{\frac{10}{3}})+C=\frac{1}{8}(1+x^3{)}^{\frac{8}{3}}-\frac{1}{5}(1+x^3{)}^{\frac{5}{3}}+C$