Articles of integration

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 […]

meaning of integration

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 […]

An integral involving a the dilogarithmic function

Consider a following definite integral: \begin{equation} \phi_a(x):=\int\limits_0^x \frac{Li_2(\xi)}{\xi+a} d \xi \end{equation} By using the integral representation of the dilogarithmic function and swapping the integration order we have shown that our integral satisfies the following functional equation: \begin{equation} \phi_a(x)= \text{Li}_3\left(\frac{a+x}{a+1}\right)-\text{Li}_3\left(\frac{a}{a+1}\right) -2 \text{Li}_3\left(\frac{1}{a+1}\right)+2 \text{Li}_2\left(\frac{1}{a+1}\right) \log \left(\frac{1}{a+1}\right)+ \text{Li}_2(-a) \log \left(\frac{a+x}{a}\right)+\text{Li}_2\left(\frac{a}{a+1}\right) \log \left(\frac{1}{a+1}\right)+\log \left(\frac{a}{a+1}\right) \log ^2\left(\frac{1}{a+1}\right)+2 \zeta (3) -\phi_{-\frac{a+x}{a}}(\frac{x+a}{1+a}) […]

Integrating the intersection of two cylinders $x^2+y^2=1$ and $y^2+z^2=1$.

Consider two intersecting cylinders. I know the regular way to do this is: $$ \int_{-1}^{1} \left(2\sqrt{1-2x^2}\right)^2dx = \frac{16}{3} $$ This methods integrates the square sides of the solid that you get However, is it just as viable to flip the picture 90° (have the yellow tube going up) and trying to integrate then? You will […]

Integration of $F(\sum_k x_k)$ over positive orthant

Problem Suppose we integrate some function $F\left(\sum\limits_{k=1}^n x_k\right)$ over the positive orthant $[0,\infty)^n$. Show that this this is proportional to the integral $\int\limits_0^\infty s^{n-1}F(s)\,ds$. What is the constant of proportionality? If this is a well-known result, a reference would be appreciated. Motivation Suppose I want to solve the inhomogeneous 1st-order ODE $(1-D)f(x)=g(x)$. Then $$D(e^{-x} f)=e^{-x}(f(x)-f'(x))=e^{-x}(1-D)f(x)=e^{-x}g(x),$$ […]