Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} e^{iz^2}\mathrm{d}z \right| $$ tends to zero as $R \to \infty$. By using the ML-inequality one has $$ \left| \int_{\gamma_2} e^{iz^2}\mathrm{d}z […]

$$\int_{0}^{\pi/2}{1+2\cos{(2x)}\ln{(\tan{x})}\over 1+\tan{x}}\mathrm dx=-{\pi\over 4}\tag1$$ Recall: $cos(2x)=\cos^2{x}-\sin^2{x}$ $$\int_{0}^{\infty}{(1+\ln{\tan^2{x}})\cos^2{x}+(1-\ln{\tan^2{x}})\sin^2{x}\over 1+\tan{x}}\mathrm dx$$ Enforcing $u=\tan^2{x}$ then $du=2\tan{x}\sec^2{x}dx$ Recall: $1+\tan^2{x}=\sec^2{x}$ and $1+\cot^2{x}=\csc^2{x}$ $${1\over2}\int_{0}^{\infty}{1+\ln{u}+u(1-\ln{u})\over 1+u}\cdot{\mathrm du\over u^{1/2}+u}$$ I am stuck! Help to prove $(1)$

This question already has an answer here: how to solve $\int_{-\infty}^\infty e^{-x^2-x{\tau}} \cdot x\ dx$? [duplicate] 1 answer

Prove that if $f$ is continuous on $[a,b]$ and $g$ is bounded variation on $[a,b],$ then $$\vert\int_a^bfdg\vert\le [sup_{a\le t \le b} \vert f(t) \vert] V_{[a,b]}g$$ Proof: As f is continuous on [a,b] and g is BV([a,b]) then f is riemann-stieltjes integrable, i.e. $f\in R(g)$. But I don’t know how to prove $\vert\int_a^bfdg\vert\le [sup_{a\le t \le […]

Define the following integral with $n$ an integer greater than $1$: $$I_{n}=\int_{0}^{1}\frac{e^t}{(1+t)^n}dt.$$ Is it true that for all $n \geq 2$, $$ \frac{1}{n-1}\left(1-\frac{1}{2^{n-1}}\right)\leq I_{n} \leq \frac{e}{n-1}\left(1-\frac{1}{2^{n-1}}\right)?$$

Let $A$ be the exact area over $[a,b]$ under $y=f(x)$. If $f(x) \geq 0$ (positive), and increasing, then $\forall x \in [a,b]$, Left Riemann Sum $\leq$ A $\leq$ Right Riemann Sum. How do I prove this? I don’t know where to start

I am looking for a technique to solve an indefinite integral of $$ \int \frac{dx}{\sqrt{ax^3+bx^2+cx+f}} $$ I honestly have no idea where to start with this and I cannot find anything like this in an integral table. Actually I was thinking maybe taylor expand? Thank you

I have read this thread and I found in some comments the above named equality. I couldn’t follow the transformation, which are done to get from the left to the right side at that point in particular. Can someone help me and show how it’s done?

I’m having a problem converting $\int\limits_1^2 \int\limits_0^ \sqrt{2x-x^2} \frac{1}{(x^2+y^2)^2} dy dx $ to polar coordinates. I drew the graph using my calculator, which looked like half a circle on the x axis. I know that $\frac{1}{(x^2+y^2)^2} dydx$ turns to $ \frac{r}{(r^2)^2}drd\theta$, which would be $ \frac{1}{r^3}$ The region of integration in $\theta $ is I […]

I read that integration is the opposite of differentiation AND at the same time is a summation process to find the area under a curve. But I can’t understand how these things combine together and actually an integral can be the same time those two things. If the integration is the opposite of differentiation, then […]

Intereting Posts

A real vector space is an inner product space if every two dimensional subspace is an inner product space ?
Zorn's Lemma $\equiv$ Axiom of Choice
Length of the tensor product of two modules
What does it mean to induce a topology?
From Gravity Equation-of-Motion to General Solution in Polar Coordinates
Proving that a linear isometry on $\mathbb{R}^{n}$ is an orthogonal matrix
trigonometry with alternative parametrizations of the circle
$\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$
Is the algebraic norm of an euclidean integer ring also an euclidean domain norm?
Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$
What are the continuous automorphisms of $\Bbb T$?
Are all uncountable infinities greater than all countable infinities? Are some uncountable infinities greater than other uncountable infinities?
Sum of the infinite series $\frac16+\frac{5}{6\cdot 12} + \frac{5\cdot8}{6\cdot12\cdot18} + \dots$
equation $\frac{\frac{1}{x^2}-\frac{1}{y^2}}{\frac{1}{x^2}+\frac{1}{y^2}}$
pullback of 1 form.