Show $\frac{2}{\pi} \int_{0}^{2\pi} f(e^{i\theta})\cos^2(\frac{\theta}{2})d\theta = 2f(0)+f'(0)$ when $f\in A(|z|<1+\varepsilon)$.
4064
An equality involving the mean value property
130058
Evaluate $\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt$
118701
Evaluate $\int_{0}^{\frac{\pi }{2}}\frac{x dx}{\sin x +\cos x}$
36701
$\displaystyle\int_{0}^{2\pi}\frac{1}{a+ \cos\theta}\,d\theta$
129892
show $\int_0^{\infty} \frac{x\sin ax}{x^2+t^2}dx = \frac{\pi}{2}e^{-at}$
4287
Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$
147759
Integral $ \dfrac { \int_0^{\pi/2} (\sin x)^{\sqrt 2 + 1} dx} { \int_0^{\pi/2} (\sin x)^{\sqrt 2 - 1} dx} $
62450
Computing $ \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$.
153318
Calculating the following integral using complex analysis: $\int_{0}^{\pi}e^{a\cos(\theta)}\cos(a\sin(\theta))\, d\theta$
10675
Proving that $\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^2dt=\sum_{n=0}^{\infty}|a_n|^2r^{2n}$
83703
Show that $\int_{0}^{\infty}|\frac{\sin x}{x}|\,dx=\infty $
69803
An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$
79044
Using the residue theorem, calculate $\int_{0}^{2\pi}\frac{1}{1-2a\cos{\theta}+a^2}d\theta$
109785
show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$
145496
Show $\lim_{h \to \ 0} \frac{f(x + 2h) - 2f(x+h) + f(x)}{h^{2}} = f''(x)$ Proof verification
139453
Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$
152336
Any hint to solve given integral $\int_0^{2{\pi}}{{d\theta}\over{a^2\cos^2\theta+b^2\sin^2\theta}}$?