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 […]

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}) […]

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 […]

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 $$D(e^{-x} f)=e^{-x}(f(x)-f'(x))=e^{-x}(1-D)f(x)=e^{-x}g(x),$$ […]

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