What will be the definite integration from 0 to inf of the cosine function  $$\int_0^\infty\cos(r)\,\mathrm dr$$
122108
What will be the formula for the following:  $$\int_0^\infty \cos(x)dx.$$
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...
109003
What is the value of $$\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt?$$
43485
Calculate the integral$$ \int_0^\infty \frac{x \sin rx }{a^2+x^2} dx=\frac{1}{2}\int_{-\infty}^\infty \frac{x \sin rx }{a^2+x^2} dx,\quad a,r \in \mathbb{R}. $$  Edit: I was able to solve the integral using complex analysis, and now I want to try and solve it using only real analysis techniques.
89151
I am practicing calculus section of GRE Math Subject test, and I can't figure out a way to do the following integral: $$\int_0^\infty \frac{\cos (ax) - \cos (bx)}{x}dx.$$  I absolutely have no clue how to do this. Can someone show me the solution explicitly? Thank you.
30747
How can I evaluate$$\int_{-\infty}^\infty \frac{\cos x}{\cosh x}\,\mathrm dx\text{ and }\int_0^\infty\frac{\sin x}{e^x-1}\,\mathrm dx.$$ Thanks in advance.
105345
I want to know how much is the integral $\int_{0}^{L}\sin(nx)\cos(mx)dx$ when $m=n$ and in the case when $m\neq n$. I know the orthogonality lemma for the other cases, but not for this one.
29793
Assmue that $f(x)\in C(0,+\infty)$ and for any $x&gt;0$ have $$\int_{x}^{x^2}f(t)dt=\int_{1}^{x}f(t)dt$$  Find this $f$  My try: we have $$2xf(x^2)-f(x)=f(x)\Longrightarrow xf(x^2)=f(x),\forall x&gt;0$$ and then I can't works
60368
Let $a&lt;1$ be a positive constant. How can I compute the following integral?  $$ \int_{0}^{2\pi} \frac{1}{1 + a \, \, \cos(x)} dx  $$
52340
I have tried the $u$ substitution yet I could not move forwards on this problem. Could you calculate in closed-form this integral? $$\int_{0}^{\pi/2}\bigg\{\csc(x)\bigg\}\mathrm{dx}$$
101093
Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$
109343
I want to compute the following integral to show that  $$ \int_0^\infty \left(\frac{1}{x}\right)^{\ln(x)}\,\mathrm{d}x=\mathrm{e}^{\frac{1}{4}}\sqrt{\pi}.$$ I have no idea where the $\sqrt{\pi}$ comes from.     Since the indefinite integral can't be evaluated with elementary functions I'm stuck.   I am just curious.
27368
Confirm that $\int_{0}^{\infty}t^{-1}\sin t dt=\pi/2$.  The guide book I am using gives the following help:  Consider $\int_{\gamma}z^{-1}e^{iz}dz$, where for $0&lt;s&lt;r&lt;\infty$ the contour of integration is described by $\gamma=[s,r]+\gamma_r+[-r,-s]-\gamma_s$, with $\gamma_r(t)=re^{it}$ and $\gamma_s(t)=se^{it}$ on $[0,\pi]$. Recall Exercise IV$.4.20.$  Exercise IV$.4.20.$ For $r$ with $0&lt;r&lt;\infty$ let $I(r)=\int_{\gamma_r}z^{-1}e^{iz}dz$ with $\gamma_r(t)=re^{it}$ on $[0,\pi]$. Show that $I(r)\rightarrow 0$ as $r\rightarrow \infty$ and also that $I(r)\rightarrow \pi i$ as $r\rightarrow 0$  Using the hint, I know that $\int_{\gamma}z^{-1}e^{iz}dz=0$ for the Cauchy theorem, with which $\int_{[s,r]}z^{-1}e^{iz}dz+\int_{\gamma_r}z^{-1}e^{iz}dz+\int_{[-r,-s]}z^{-1}e^{iz}dz-\int_{\gamma_s}z^{-1}e^{iz}dz=0$, but I do not know what else to do here, could someone help me please? Thank you very much.
111409
Evaluate: $$I=\int\limits_{0}^{\frac\pi2}\ln\frac{(1+\sin x)^{1+\cos x}}{1+\cos x}dx$$
92455
$$\int_{-\infty}^{\infty} \frac{\cos(αx)}{(x^2+1)(x^2+4)} \mathrm dx. $$  I am not sure how to solve this question. Can anyone help me to approach this problem. Thanks.
9190
I'm working on the Fourier transform, but I don't know how to evaluate the integral:  $$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$
2645
How to prove  $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$