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If I=int(dx)/(x^(4)sqrt(a^(2)+x^(2))), ...

If `I=int(dx)/(x^(4)sqrt(a^(2)+x^(2)))`, then I equals

A

`(1)/(a^(4))[(1)/(x)sqrt(a^(2)+x^(2))-(1)/(3x^(2))sqrt(a^(2)+x^(2))]+c`

B

`(1)/(a^(4))[(1)/(x)sqrt(a^(2)+x^(2))-(1)/(2sqrtx)(a^(2)+x^(2))^(3//2)]+c`

C

`(1)/(a^(4))[(1)/(x)sqrt(a^(2)+x^(2))-(1)/(3x^(3))(a^(2)+x^(2))^(3//2)]+c`

D

`(1)/(a^(4))[(1)/(x)sqrt(a^(2)+x^(2))-(1)/(3x^(3))sqrt(a^(2)+x^(2))]+c`

Text Solution

Verified by Experts

The correct Answer is:
C
`x^(4)sqrt(a^(2)+x^(2))=x^(5)sqrt(((a)/(x))^(2)+1)`
`therefore" "I=int((dx)/(x^(3)))/(x^(2)sqrt(((a)/(x))^(2)+1))`
Let `(a^(2))/(x^(2))=t, " so "(-2a^(2))/(x^(3))dx=dt`
`therefore" "I=-(1)/(2a^(4))int(t)/(sqrt(t+1))dt=-(1)/(2a^(4))int(t+1-1)/(sqrt(t+1))dt`
`=(1)/(a^(4))[sqrt(t+1)-(1)/(3)(t+1)^(3//2)]+c`
`=(1)/(a^(4))[(sqrt(a^(2)+x^(2)))/(x)-(1)/(3x^(3))(a^(2)+x^(2))^(3//2)]+c`
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