Articles of integration

Understanding Riemann sums

I am trying to understand Riemann sums. As far as I can understand, we have $\Delta x$ which is $\frac{b-a}{n}$ and $n$ is the number of subintervals I want to divide my function between $a$ and $b$. But I still don’t understand how to calculate $C_i$. For example I have a function in my book […]

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

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 […]

Help to prove that $\int_{0}^{\pi/2}{1+2\cos{(2x)}\ln{(\tan{x})}\over 1+\tan{x}}\mathrm dx=-{\pi\over 4}$

$$\int_{0}^{\pi/2}{1+2\cos{(2x)}\ln{(\tan{x})}\over 1+\tan{x}}\mathrm dx=-{\pi\over 4}\tag1$$ 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 $(1)$

Integrate $\int_{-\infty}^\infty xe^{-\alpha x^2+\beta x}dx$

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

Riemann-stieltjes integral and the supremum of f

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 […]

Inequality for integral

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 $n \geq 2$, $$ \frac{1}{n-1}\left(1-\frac{1}{2^{n-1}}\right)\leq I_{n} \leq \frac{e}{n-1}\left(1-\frac{1}{2^{n-1}}\right)?$$

How to prove Left Riemann Sum is underestimate and Right Riemann sum is overestimate?

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

looking for a technique to solve an indefinite integral of one over the square root of a cubic polynomial

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

Evaluate $ \int_0^1 \sum_{k=0}^\infty (-x^4)^k dx = \int_0^1 \frac{dx}{1+x^4} $

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?

Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates

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 […]