If a definite integral represents the area under the curve, why is $\int_{0}^{\pi/2} \ln|\sin x| \, dx = -\pi \ln(2)/2$?
55341
The negative integral meaning
25797
Integral of $ x \ln( \sin (x))$ from 0 to $ \pi $
105423
Why doesn't every integral from 0 to $2\pi$ equal zero?
91333
How to find $\int_{0}^{1}\dfrac{\ln^2{x}\ln^2{(1-x)}}{2-x}dx$
135653
An integral$\frac{1}{2\pi}\int_0^{2\pi}\log|\exp(i\theta)-a|\text{d}\theta=0$ which I can calculate but can't understand it.
125506
Why can I get two different results when calculating $\int \ln(z) \,dz$?
90245
Computing $\int^1_0 \frac{p^2-1}{\text{ln } p}dp$
62840
Seemingly Simple Integral $\int_0^1\frac{x^2\ln x}{\sqrt{1-x^2}}dx$.
20016
How to integrate $\int_0^{\pi/2}\frac{\sin^nx}{\sin^nx+\cos^nx}dx$?
126654
What is the definite integral $\int_{-1}^1 \frac{1}{x} dx$ equal to?
8358
Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $
6892
Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?
61585
Why is area of a surface of revolution integral $2\pi y~ds$? not '$dx$'?
118330
How do we know what the integral of $\sin (x)$ is?
123899
$\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}$
47779
Why does this integral have a complex component?
83966
Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?