Solve the integral: ∫02dxx(x−1)

compagnia04

compagnia04

Answered

2022-01-05

Solve the integral:
02dxx(x1)

Answer & Explanation

encolatgehu

encolatgehu

Expert

2022-01-06Added 27 answers

You can break up the integral at any point or points you like. In this case, you could break it up into three integrals: pick a point c strictly between 0 and 1, and consider:
02dxx(x1)=0cdxx(x1)+c1dxx(x1)+12dxx(x1)
=lima0+acdxx(x1)+limb1cbdxx(x1)+limd2dxx(x1)
The original improper integral exists if and only if each of the three improper integrals exist.
Jenny Bolton

Jenny Bolton

Expert

2022-01-07Added 32 answers

Consider the change of variables t=x. It transforms the interval [0,2] into [0,2] and removes one of the problem zeroes. It is easy to see from the resulting expression that the integral diverges.
karton

karton

Expert

2022-01-11Added 439 answers

Try to solve the integral. You'll find whether it converges. For example, dxx(1x) with ϵ>0. Later on, you can study the limit ϵ0+. In this way, you ''kill two birds with a one shot''.

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