Calculus/Calculus- Definite Intregrals
I have 2 questions from a book at a point where L’Hopital’s Rule, Squeeze Theorem etc. have not been discussed and limits (A) and (B) as given below are to be evaluated by simple methods like algebraic simplification etc.
1. Int. of (xlogx)dx from 0 to 1.
Indefinite Int. comes to [x^2/2*logx-x^2/4].
Applying limits, Def. Int. =(1/2log1-1/4) – [lim as x->0 of(x^2/2*logx-0)] = -1/4, which is the required answer, taking limit of (x^2/2*logx)=0 as x->0 ……(A).
2. Int. of (x^2*e^-x)dx from 0 to infinity.
Indefinite Int. comes to [e^-x(-x^2-2x-2)].
Applying limits, Def. Int.= – [lim as x->0 of [ (-x^2-2x-2) /e^x *]
= lim as x->infinity, of [(-x^2-2x-2)/ e^x]2 – (-2)= 2, which is the required answer, taking limit of [1/e^x(-x^2-2x-2]=0 as x->infinity ……(B).
Is it possible to deduce these limits (A) and (B) without L’Hopital’s Rule, Squeeze Theorem etc. ?
1) Those questions are not "definite integral" questions, they belong to "Improper Integrals". the integrand (function between the integral sign and dx) is not define somewhere between the interval of the integral limits. So it inevitably will become a "Limit" question are evaluating the integration and substituting the lower and upper values (of the definite integrals)
2) Usually if the final function comes up to be just e^(-x) or ln(x), ln(1-x), 1/e^(2x) etc, then the limits are very easy to evaluate (most calculus teacher won't even need you to show work, or tell them Squeeze Theorem is used, the answer is usually either 0 or infinity or negative infinity). But if it is a combination of exp, ln and polynomial (or worse trigonometric function), then L'Hopital's rule become very useful.
3) When you said "algebraic simplification", I know what it means, but what does "etc" include (or exclude)? It is very dangerous to use a word as ambiguous as "etc" in Mathematics. Would you allow "Residue Theory", and "Cauchy-Riemann" method or "Contour Integral" as part of "etc"? I know the answer is obviously not because those terms you'll probably not hear them or learn them, but in the history of mathematics, some unguided famous mathematicians like "Ramanujan", used method that were easy, useful but even top mathematicians never thought of.
4) Typically, most people will answer you with a simple "NO". But my answer is what I'm giving you now in 4 parts.