if x=phi(t) and intf(x)dx=F(x) then intf...

if `x=phi(t)` and `intf(x)dx=F(x)` then `intf(phi(t))phi'(t)dt=` (A) `phi(x)` (B) `F(t)` (C) `F(x)` (D) `F^(')(x)`

Text Solution

Verified by Experts

`x=phi(t)` is differentiable function of t.
`therefore (dx)/(dt)=phi(t)`
`Let intf(x)dx=F(x) therefore (d)/(dx)[F(x)[=f(x)`
`therefore` by the chain rule.
`(d)/(dt)[F(x)]=(d)/(dx)[F(x)].(dx)/(dt)`
`=f(x).(dx)/(dt)=f[phi (t)].phi(t)`
`therefore` by the definition of integral,
`F(x)=intf[phi(t)].phi(t)dt`
`therefore intf(x)dx=intf[phi(t).phi'(t)dt.`

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