- #1
Eclair_de_XII
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- 91
- TL;DR Summary
- I've read somewhere that if a function is integrable over some domain, then a scalar multiple of that function is also integrable over the same domain. Moreover, the integral of a scalar multiple of a function is the scalar multiple of the integral of that function. But when I think about the integral of the function ##f(x)=\frac{1}{x}##, I start to get confused.
Let's take an integral##−\int_1^e\frac{dx}{x}##. On one hand, this is equal to ##-\ln(x)|_1^e##. But on the other, ##−\int_1^e\frac{dx}{x}=\int_1^e\frac{dx}{-x}##. If I assume that the integral of this is ##\ln(-x)|_1^e##, then I'd be really stupid since ##\ln## is not even defined over the negative real numbers. I am very sure I am misunderstanding something very important in these statements I have cited. Or maybe I forgot some key statements that go along with these. Either way, I'm confused, and it's been bothering me all day. Can someone help me point out my mistakes? Thanks.