InterviewSolution
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InterviewSolution
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int _(-1)^(1)e^(x) dx |
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Answer» Solution :we knowthat `overset(a)underset(b)(int)f(x)DX=underset(h TO0)("lim") h[f(a)+f(a+h)` `+f(a+2H)+....+f{a+(n-1)}h]` wherenh=b-a `" GIVEN" overset(1)underset(-1)(int) e^(x) dx` `" Here "a=-1 ,b=1 " and " NH=b-a=2` `" and "f(x) =e^(x)` `:. overset(1)underset(-1)(int) e^(x) dx= underset(h to 0)(" lim") [e^(-1)+e^((-1+h))+e^((-1+2h))` `+.....+e^({-1+(n-1)h})]` `=underset(h to0)("lim") he^(-1)[1+e^(h)+e^(2h)+......+e^((n-1)h)]` `=underset(h to0)("lim") (h)/(e) ({(e^(h))^(n)-1})/(e^(n)-1)=(1)/(e)underset(h to0)("lim") (e^(hn)-1)/((e^(h)-1)/(h))` `(:. |e^(h)|gt 1)` `=(1)/(e)underset(h to0)("lim") (e(2)-1)/((e^(h)-1)/(h))=(1)/(e)((e^(2)-1)/(1))` `"(":' underset(h to0)("lim") (e^(x)-1)/(x)=1 " and nh"=2")"` `=(e^(2)-1)/(e)=e-(1)/(e)` |
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