`"If "int x^(5)(1+x^(3))^(2//3)dx=A(1+x^(3))^(8//3)+B(1+x^(3))^(5//3)+c, " then " `find A and B

Options

A. `A=(1)/(4),B=(1)/(5)`
B. `A=(1)/(8),B=-(1)/(5)`
C. `A=-(1)/(8),B=(1)/(5)`
D. non of these

Answer

B

Explanation

`"Here, " int x^(5)(1+x^(3))^(2//3)dx`
`"Let " 1+x^(3)=t^(2) " and " 3x^(2)dx=2tdt `
`:. int x^(5)(1+x^(3))^(2//3)dx =int x^(3)(1+x^(3))^(2//3)x^(2)dx `
`=int(t^(2)-1)(t^(2))^(2//3)x^(2)dx `
`=(2)/(3)int(t^(2)-1)t^(7//3)dt`
`=(2)/(3)int(t^(13//3)-t^(7//3))dt `
`=(2)/(3){(3)/(16)t^(16//3)-(3)/(10)t^(10//3)}+C `
`=(1)/(8)(1+x^(3))^(8//3)-(1)/(5)(1+x^(3))^(5//3)+C`