`int_(-1)^(1)e^(x)dx`

Explanation

हम जानते हैं
`int_(1)^(b)f(x)dx=underset(hrarr0)(lim)h[f(a)+f(a+h)+f(a+2h)+…+f(a+(n-1)}h]`
जहाँ, `nh=b-a`
दिया है, `int_(-1)^(1)e^(x)dx`
यहाँ,`" "a=-1, b=1" और "nh=b-a=2`
और `" "f(x)=e^(x)`
`therefore" "int_(-1)^(1)e^(x)dx=underset(hrarr0)(lim)h[e^(-1)+e^((-1+h))+e^((-1+2h))+....+e^({-1+(n-1)h})]`
`=underset(hrarr0)(lim)he^(-1)[1+e^(h)+e^(2h)+...+e^((n-1)h)]`
`=underset(hrarr0)(lim)(h)/(e)({(e^(h))^(n)-1})/(e^(h)-1)=(1)/(e)underset(hrarr0)(lim)(e^(hn)-1)/((e^(h)-1)/(h))`
`" "(because|e^(h)|gt1)`
`=(1)/(e)=underset(hrarr0)(lim)(e^(2)-1)/((e^(h)-1)/(h))=(1)/(e)((e^(2)-1)/(1))`
`" "(because underset(xrarr0)(lim)(e^(x)-1)/(x)=1" और "nh=2)`
`=(e^(2)-1)/(e)=e-(1)/(e)`