query: How to calculate $\int_0^1e^{i \pi x}x^x(1-x)^{(1-x)}dx$=?
positive_passages.docid: 28771
positive_passages.text: Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $
negative_passages.docid: 115089
negative_passages.text: Definite Integral: $\int_0^1\frac{x^{2} -1}{\log x}\,dx$
negative_passages.docid: 83290
negative_passages.text: How is the value of $\pi$ ( Pi ) actually calculated?
negative_passages.docid: 150809
negative_passages.text: Compute $ \int_0^1\frac{\ln^2(1+x)}{1+x^2}\, dx$
negative_passages.docid: 79361
negative_passages.text: What is the integral of 1/x?
negative_passages.docid: 147759
negative_passages.text: Integral $ \dfrac { \int_0^{\pi/2} (\sin x)^{\sqrt 2 + 1} dx} { \int_0^{\pi/2} (\sin x)^{\sqrt 2 - 1} dx} $
negative_passages.docid: 104691
negative_passages.text: Integration of $\int_{0}^{1} \frac{x^k}{1-x} dx$
negative_passages.docid: 101587
negative_passages.text: Calculating $\int_{- \infty}^{\infty} \frac{\sin x \,dx}{x+i} $
negative_passages.docid: 4833
negative_passages.text: Compute $\int_0^1\frac{f(x)}{f(x)+f(1-x)}dx$ and $\int_{-\pi}^{\pi}\frac{1}{1+e^{\sin (x)}}\,dx$?
negative_passages.docid: 84826
negative_passages.text: Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis
negative_passages.docid: 74454
negative_passages.text: Sophomore's dream: $\int_0^{1} x^{-x} \; dx = \sum_{n=1}^\infty n^{-n}$
negative_passages.docid: 27195
negative_passages.text: Evaluating $\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$
negative_passages.docid: 21464
negative_passages.text: Finding the integral $\int_0^1 \frac{x^a - 1}{\log x} dx$
negative_passages.docid: 51216
negative_passages.text: Different ways to tackle the integral $\int_0^1\sqrt\frac x{1-x}\,dx$
negative_passages.docid: 99753
negative_passages.text: How to prove $\int_0^1\frac{1-x}{(\ln x)(1+x)}\ dx=\ln\left(\frac2{\pi}\right)$?
negative_passages.docid: 42758
negative_passages.text: How to find $ \int_0^\infty \dfrac x{1+e^x}\ dx$
negative_passages.docid: 6830
negative_passages.text: Find $\int\frac1x\sqrt\frac{1-x}{1+x}\ dx$