Can we find the integral $$\int_0^{2\pi} \cos(\cos x)\,dx\;?$$
123795
Evaluate $\int \cos(\cos x)~dx$   I tried to use chain rule but failed. Can anyone help me please?
72470
I am trying to solve this integral, I think that it could be solve using the complex.   $$ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $$
61065
I want to calculate the integral  $$\int_0^{\frac{\pi}{2}} e ^{ \sin t}\, dt.$$ Can we find a primitive function for $f(t) = e ^{\sin t}$?
92399
$$\int_0^{\pi/2} \sin^a x \cos^b x \,\mathrm{d} x = \frac{\Gamma\left(\frac{1+a}{2}\right)\Gamma\left(\frac{1+b}{2}\right)}{2 \Gamma\left(1 + \frac{a + b}{2}\right)} \quad\quad\text{for } a, b &gt; -1$$ according to Mathematica.  also lists a recursive expression for the indefinite integral when $a, b &gt; 0$. My question is how to derive the explicit formula given by Mathematica (preferably without using esoteric special functions, but complex analysis is fine).
112380
$$\int_0^\pi \sqrt{4\sin^2 x - 4\sin x + 1}\,dx$$  Please help with this. I cannot do this problem in a definite way.
39298
I'd love your help with finding the sign of the following integral: $\int_{0}^{2 \pi}\frac{\sin x}{x} dx$,I know that computing it is impossible.  I tried to use integration by parts and maybe to learn about the sign of each part and conclude something but It didn't work for me.  Any suggestions?
28624
Calculate the integral for $\left|a\right|&lt;1$  $$\int_{-\pi}^{\pi} \dfrac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$$  I'm supposed to evaluate this using method of residues, but the parameter a is confusing me. I'm not even sure what contour I should use. Any help would be very appreciated.
114861
Is this correct? I thought it would be but when I entered it into wolfram alpha, I got a different answer.   $$\int (\cos^3x)(\sin^2x)dx  = \int(\cos x)(\cos^2x)(\sin^2x)dx  = \int (\cos x)(1-\sin^2x)(\sin^2x)dx.$$  let $u = \sin x$, $du = \cos xdx$  $$\int(1-u^2)u^2du = \int(u^2-u^4)du  = \frac{u^3}{3} - \frac{u^5}{5} +C$$  Plugging in back $u$, we get $\displaystyle\frac{\sin^3 x}{3} - \frac{\sin^5 x}{5}$ + C
48250
The problem: Find the value of $\displaystyle\int_0^{\pi/2} \frac{dx}{1+(\tan(x))^{\sqrt{2}}}$   I tried a few different substitutions and the closest I got to an okay looking integral is $$\int_0^\infty \frac{t^2 \, dt}{t^{1/\sqrt{2}}(1+\sqrt{t})},$$ which still looks scary. How do I approach this integral?
122986
Quick question, might end up having a simple answer, but I have here a "proof" that any integral from 0 to $2\pi$ is zero, as follows:  $$\int^{2\pi}_0f(x)dx$$  Now using u-substitution, let $u = \sin x$, so $dx = \frac{du}{\cos x}$, so:  $$\int^{0}_{0}\frac{f(x)du}{\cos x}$$  (The bounds were evaluated to zero due to the u-substituion).  However, this is zero, since the integral goes from zero to zero.  Any help as to why this is wrong would be appreciated.
37587
It's my first  post here  and I was wondering  if someone  could help me with evaluating the definite integral  $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in advance, any help would be appreciated.
91157
Question:   Compute $$\int _0 ^{2\pi} \left(\sum_{n=1}^\infty \frac {\cos(nx)}{2^n}\right)^2 dx$$   Thoughts: Tried interchanging the integral and the sum, but then the integral turned out to be zero...
12582
My problem is to integrate this expression:  $$\int_0^{2\pi}\log(1-2r\cos x +r^2)dx.$$ where $r$ is any constant in $[0,1]$.  I know the answer is zero. Can you explain you idea to me or just prove that? Maybe you will use the "Cauchy integral theorem ".
1954
$$\int \cos^4 x\,\mathrm dx  = \int (1 - \sin^2x)^2\,\mathrm dx.$$  I tried using $\cos^2(x) + \sin^2(x)$ = 1. This was on the integration by parts section of my textbook.  The integral I came out with is given me a hard time. I double checked my work and I feel I am on the right track but I feel stuck. I tried u substitution but there is no $\cos x$ to leverage with.
107906
I have to verify that     $$\int_0^\pi \ln(1+\alpha\cos(x))dx=\pi\ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$$   with $|\alpha|&lt;1$. It is my homework and don't know where to begin.
52340
I have tried the $u$ substitution yet I could not move forwards on this problem. Could you calculate in closed-form this integral? $$\int_{0}^{\pi/2}\bigg\{\csc(x)\bigg\}\mathrm{dx}$$
87553
How to figure out the following integral? I have not been able to solve it from some time. $$\int_0^1\frac{x^{2} -1}{\log x}\,dx$$