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 toA. -1B. 1C. `-(1)/(2)`D. `-(5)/(2)`

If `int x^(5)e^(-x^(2))dx = g(x)e^(-x^(2))+C`, where C is a constant o

1.	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 toA. -1B. 1C. `-(1)/(2)`D. `-(5)/(2)`
Answer» Correct Answer - D Let given integral, `I = int x^(5) e^(-x^(2)) dx` Put `x^(2) = t rArr 2xdx = dt` So, `I=(1)/(2)int t^(2)e^(-1)dt` `=(1)/(2)[(-t^(2)e^(-t))+inte^(-t)(2t)dt]" "["Integration by parts"]` `=(1)/(2)[-t^(2)e^(-t)+2t(-e^(-t))+int 2e^(-t)dt]` `=(1)/(2)[-t^(2)e^(-t)-2te^(-t)-2e^(-t)]+C` `=-(e^(-t))/(2)(t^(2)+2t+2)+C` `=-(e^(-x^(2)))/(2)(x^(4)+2x^(2)+2)+C" "[therefore t = x^(2)]" "....(i)` `therefore` It is given that, `I=int x^(5)e^(-x^(2))dx = g(x)*e^(-x^(2))+C` By Eq. (i), comparing both sides, we get `g(x)=-(1)/(2)(x^(4)+2x^(2)+2)` So, `g(-1)=-(1)/(2)(1+2+2)=-(5)/(2)`

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