Articles of integration

Showing that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges (Baby Rudin Exercise 6.9)

Motivated by Baby Rudin Exercise 6.9 I need to show that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges. My attempt: $\frac{|\cos x|}{1+x} \geq \frac{\cos^2 x}{1+x}$, and then $\int_0^\infty \frac{\cos^2 x}{1+x} \, dx + \int_0^\infty \frac{\sin^2 x}{1+x} \, dx = \int_0^\infty \frac{1}{1+x} \, dx$. Since the right integral diverges, either or both of the integrals on the […]

First-order nonlinear ordinary differential equation

How to solve this differential equation: $$x\frac{dy}{dx} = y + x\frac{e^x}{e^y}?$$ I tried to rearrange the equation to the form $f\left(\frac{y}{x}\right)$ but I couldn’t thus I couldn’t use $v = \frac{y}{x}$ to solve it.

Fourier Uniqueness Theorem: Proof?

I need this as lemma. Given the Borel space $\mathcal{B}(\mathbb{R})$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{R})\to\mathbb{C}$$ Then one has: $$\int_{-\infty}^{+\infty}e^{it\lambda}\mathrm{d}\mu(\lambda)=0\implies\mu=0$$ How can I prove this?

How to calculate this Ei(x)-involved definite integral?

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al). Finally, I decided to solve this integral by means of ” Residue Theorem “. Can anyone help me ? $$ \int_u^\infty \frac{Ei(-x)e^{-px}}{(x-\beta)}dx\,. $$ $$ p>0\ ,\ \beta>0\ \ ,\ u>0 […]

Is there another simple way to solve this integral $I=\int\frac{\sin{x}}{\sin{x}+\cos{x}}dx$?

The integral I want to find is$$I=\int\frac{\sin{x}}{\sin{x}+\cos{x}}dx$$ The way I learnt is to introduce$$J=\int\frac{\cos{x}}{\sin{x}+\cos{x}}dx$$ Then $J+I=x+C_1$ and $J-I=\ln|\sin{x}+\cos{x}|+C_2$. Is there some simple way to solve this integral $I$? For example, do not introduce other integrals? Any hints will be appreciated. Thank you.

Change of variable (translation) in complex integral

If I have a real integral, e.g. $\int f(x+2) \ dx$, I can substitute $y = x+2$, so $dy = dx$. But if my function is complex, am I still allowed to do this? In which cases I cannot apply a translation?

How to prove Raabe's Formula

This question already has an answer here: Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$ 4 answers

Relation between de Rham cohomology and integration

This question is a follow-up to When does a null integral implies that a form is exact? . As mentionned in the selected answer, given certain conditions it is possible to find an isomorphism between the top de Rham cohomology and the integral $\int_M \omega$ of the members of each equivalence class $\omega$. Moving on […]

Show that $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2}$ + Constant

I tried to do this integration by parts and got $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2} +\alpha\int x^3\mathrm{e}^{-\alpha x^2}\mathrm dx$ + constant. Where $\alpha$ is a constant. Any help will be most appreciated. Thank you.

Solving an integral (with substitution?)

For a physical problem I have to solve $\sqrt{\frac{m}{2E}}\int_0^{2\pi /a}\frac{1}{(1-\frac{U}{E} \tan^2(ax))^{1/2}}dx $ I already tried substituting $1-\frac{U}{E}\tan^2(ax)$ and $\frac{U}{E}\tan^2(ax)$ since $\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin(x)$ but my problem is that $dx$ changes to something with $\cos^2(ax)$, thus making the integral not easier. Anyone got a hint? EDIT: The physical problem is to calculate the oscillating […]