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If int x ^ 5 e ^ ( - x ^2 ) dx = g...

If ` int x ^ 5 e ^ ( - x ^2 ) dx = g (x) e ^( -x ^ 2 ) + c, ` where c is a constant of integration , then ` g (-1) ` is equal to :

A

1

B

`-1`

C

`-(5)/(2)`

D

`-(1)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `I = int (x^(2))^(2) e^(-x^(2)) x dx = g(x) e^(-x^(2)) + c`
Let `x^(2) = t, 2xdx = dt = (1)/(2) int t^(2) e^(-t) dt = (1)/(2) [-e^(-t)t^(2) + int 2t e^(-t) dt] = (1)/(2) [-e^(-t) t^(2) -2t e^(-t) + int 2e^(-t) dt]`
`= (1)/(2) [-e^(-t) .t^(2) - 2te^(-t) - 2e^(-t)] =e^(-t) [(-t^(2) -2t -2)/(2)] = e^(-x^(2)) [(-x^(4)-2x^(2) -2)/(2)]`
`g(x) = (-x^(4) -2x^(2) -2)/(2) rArr g(-1) = (-1 -2-2)/(2) = -(5)/(2)`
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