3
votes
1answer
45 views

Show the integrals are alike without explicit computation.

I saw a couple of striking integrals which are $\hspace{5em} \displaystyle \color{black}{\displaystyle \int_a^b \frac{x\,\mathrm{d}x}{\sqrt{(x^2-a^2)(b^2-x^2)}} }$$\displaystyle\ =\ $$\displaystyle ...
1
vote
1answer
42 views

Is exponential even?

My book presents the following simplification: $$A(w) = \frac{1}{\pi}\int_{-\infty}^{\infty} e^{-kx}cos(wx)dx = \frac{2}{\pi}\int_{0}^{\infty} e^{-kx}cos(wx)dx$$ But I can't understand why, since ...
4
votes
3answers
650 views

Why isn't an odd improper integral equal to zero

My calculus book says that the integral of $\frac1x$ cannot cross zero. Now it seems obvious that because of symmetry, there will always be an interval whose integrals are equal in magnitude and ...