I attempted to integrate the following function from a practice problem in my Calculus textbook:
$$\int_{0}^{\frac{\pi}{2}}{\frac{1}{1+\tan^\sqrt{2}(x)}} \ {\rm d}x$$
I failed to find an indefinite integral, and I am assuming getting an indefinite integral is simply impossible. Using Wolfram|Al...
Can this integral be solved with contour integral or by some application of Residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catlan constant}$$
It has two poles at $\pm i$ and branch point of $-1$ while the integral is to be evaluated from $0\to \infty$....
So far, I've unlocked one of the secret hats. I'm wondering what other secret hats are out there and how to earn them?
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
It is the first of the polygamma functions.
== Relation to harmonic numbers ==
The digamma function, often denoted also as ψ0(x), ψ0(x) or (after the shape of the archaic Greek letter Ϝ digamma), is related to the harmonic numbers in that
where Hn is the n-th harmonic number, and γ is the Euler-Mascheroni constant. For half-integer values, it may be expressed as
== Integral representations ==
If the real part of x is positive then the digamma function has the following integral representation...
Brown is the color of dark wood or rich soil. It is a composite color; in printing or painting, it is made by combining red, black and yellow, or red, yellow and blue. In the RGB color model used to make colors on television screens and computer monitors, it is made by combining red and green, in specific proportions. The color is seen widely in nature, in wood, soil, and human hair color, eye color and skin pigmentation. In Europe and the United States, it is the color most often associated with plainness, humility, the rustic, and poverty. It is also, according to public opinion surveys in Europe...
