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 […]

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 […]

Intereting Posts

How to find the determinant of this $5 \times 5$ matrix?
How does one evaluate $\lim\limits _{n\to \infty }\left(\prod_{x=2}^{n}\frac{x^3-1}{x^3+1}\right)$?
Multiplicative Inverse of a Power Series
A *finite* first order theory whose finite models are exactly the $\Bbb F_p$?
Power method for finding all eigenvectors
Green's Function ODE
What if the Euler Lagrange equation yields a 'trivial' answer
Do different methods of calculating fractional derivatives have to be equal?
What is the categorical diagram for the tensor product?
Sequence of random variables with stopping rule
Bijection between sets of ideals
How to show that these groups are isomorphic?
Can I think of Algebra like this?
Triangle problem – finding the angle
Simplifying Ramanujan-type Nested Radicals