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int sqrt(x)(1+x^(1//3))^(4)dx " is equal...

`int sqrt(x)(1+x^(1//3))^(4)dx " is equal to "`

A

`2{x^(2//3)+(4)/(11)x^(11//6)+(6)/(13)x^(13//6)+(4)/(15)x^(5//2)+(1)/(17)x^(17//6)}+c`

B

`6{x^(2//3)-(4)/(11)x^(11//6)+(6)/(13)x^(13//6)-(4)/(15)x^(5//2)+(1)/(17)x^(17//6)}+c`

C

`6{(1)/(9)x^(3//2)+(4)/(11)x^(11//6)+(6)/(13)x^(13//6)+(4)/(15)x^(5//2)+(1)/(17)x^(17//6)}+c`

D

non of these

Text Solution

Verified by Experts

The correct Answer is:
C
`"Let " x=t^(6) " or " dx=6t^(5)dt`
`:. I=int t^(3)(1+t^(2))^(4)6t^(5)dt `
`=6int t^(8)(1+4t^(2)+6t^(4)+4t^(6)+t^(8))dt`
`=6int(t^(8)+4t^(10)+6t^(12)+4t^(14)+t^(16))dt`
`=6{t^(9)/(9)+(4t^(11))/(11)+(6t^(13))/(13)+(4t^(15))/(15)+t^(17)/(17)}+C`
`=6{(1)/(9)x^(3//2)+(4)/(11)x^(11//6)+(6)/(13)x^(13//6)+(4)/(15)x^(5//2)+(1)/(17)x^(17//6)}+C`
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