Intereting Posts

Is XOR a combination of AND and NOT operators?
Supplement to Herstein's Topics in Algebra
Period of 3 implies chaos
Definite integral: $\displaystyle\int^{4}_0 (16-x^2)^{\frac{3}{2}} dx$
How to prove these two random variables are independent?
A continued fraction for $\sqrt{2}\Bigg({e^\sqrt{2}-1 \above 1.5pt e^\sqrt{2}+1 }\Bigg)$
Factor $10^n – 1$
Find a formula for all the points on the hyperbola $x^2 – y^2 = 1$? whose coordinates are rational numbers.
Combinatorial proof of a Stirling number identity.
Prove that $\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$ converges and find its sum
$Tf=\sum\limits_{n=1}^\infty f(n)x_n$ is surjective from $\ell^1$ to a separable Banach space
prove this :$\sum_1^{100} A_i = \sum_0^{99} C_i $
proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
Distinguishable/indistinguishable objects and distinguishable/indistinguishable boxes
Why is $e^{\pi \sqrt{163}}$ almost an integer?

Calculate the integral

$$\int \ln (\sin x) \, dx.$$

- Find $\lim_{n\to\infty}\sqrt{6}^{\ n}\underbrace{\sqrt{3-\sqrt{6+\sqrt{6+\dotsb+\sqrt{6}}}}}_{n\text{ square root signs}}$
- Is $0$ an Infinitesimal?
- A continuously differentiable function with vanishing determinant is non-injective?
- I need assistance in integrating $ \frac{x \sin x}{1+(\cos x)^2}$
- Puiseux Series?
- A question on the Stirling approximation, and $\log(n!)$
- If $I_n=\sqrt{\int _a^b f^n(x) dx}$,for $n\ge 1$.Find with proof $\lim _{n \to \infty}I_n$
- Inequality constraints in calculus of variations
- When can we plug in values in a limit?
- Why I am getting different answer?

Consider the following.

\begin{align}

I &= \int \ln(\sin(x)) \ dx

\end{align}

can be evaluated by integration by parts and leads to

\begin{align}

I &= x \ln(\sin(x)) – \int x \ \cot(x) \ dx \\

&= x \ln(\sin(x)) -x \ln(1 – e^{ix}) – \frac{i}{2} \left( x^2 + \operatorname{Li}_2(e^{2ix}) \right)

\end{align}

where $i =\sqrt{-1}$ and $\operatorname{Li}_2(z)$ is the dilogarithm function. It is of note that

\begin{align}

\int_0^{\pi/2} \ln(\sin(x)) \ dx = \frac{\pi}{2} \ln(2).

\end{align}

- Sum of real numbers that multiply to 1
- How do I find the matrix with respect to a different basis?
- Is there any formula of monadic second-order logic that is only satisfied by an infinite set?
- $F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$
- $|G|>2$ implies $G$ has non trivial automorphism
- Geometric justification for the prime spectrum and “generic points”
- Who is a Math Historian?
- Coupon collector without replacement
- Elementary proof that $3$ is a primitive root of a Fermat prime?
- Orientability of $m\times n$ matrices with rank $r$
- No function that is continuous at all rational points and discontinuous at irrational points.
- Qualification of a Universal Quantification
- A conjecture about integer polynomials and primes
- Why is the probability that a continuous random variable takes a specific value zero?
- Help with System of nonlinear equations!