5
votes
0answers
107 views
2
votes
1answer
41 views

Intuition concerning Riemann Sums

I have just started learning integrals, and I want to know the following: In the definition of a riemann integral, it states that the interval that the integral is to be evaluated, is partitioned ...
2
votes
3answers
81 views

Evaluate $\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} $

$$\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} dx$$ My approach is to calc $$\int_{1}^{X} \frac{\ln{(2x-1)}}{x^2} dx$$ and then take the limit for the answer when $X \rightarrow \infty$ However, I must ...
0
votes
3answers
32 views

Integral of a function with two parts (piecewise defined)

The function has 2 parts: $$f(x) = \begin{cases} -\sin x & x \le 0 \\ 2x & x > 0\end{cases}$$ I need to calculate the integral between $-\pi$ and $2$. So is the answer is an integral ...
0
votes
2answers
17 views

Average Value - Graphs

long method: Determine an equation for each and solve using average value formula alternative methods? How could you prove the average value to be C over an interval [a,b] if you are given a ...
0
votes
2answers
37 views

How to use trigonometric substitution to compute this definite integral?

I have searched for a similar question on stack exchange but could not find one. The definite integral: $\large\int_0^1 \frac{x^4}{\sqrt{25-x^2}}$ I realize that I need to use $x = 5\sinθ$ in the ...
1
vote
1answer
45 views

Determining the infinite limit of a Riemann' sum

I need to evaluate the following $f(x)=x^2 + 2x - 5$, on $[1,4]$, by using the riemans sum and limiting it to infinity. I have set up everything $\Delta x=\frac{3}{n}$. $x_i=1+\frac{3i}{n}$, I would ...
2
votes
1answer
86 views

Examples of pairs of difficult integrals

I’m looking for pairs of difficult definite integrals that are linked algebraically on a certain field without known change of variable or integration by parts from one integral to the other. Here a ...
1
vote
1answer
121 views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

How do you arrive at the result $$I=\displaystyle\int_{\mathbb{R^{+}}} \dfrac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\dfrac{\pi}{32} (3\pi-4)^2\ ?$$ Wolfram Alpha agrees numerically. I tried ...
0
votes
2answers
105 views

Improper integral of $\frac{\ln x}x$

Find $$\int_e^{\infty}\frac{\ln x}{x}\ dx$$ $A.\ \dfrac12$ $B.\ \dfrac{e^2}{2}$ $C.\ \dfrac{\ln(2e)}{2}$ $D.$ DNE (Does not exist) I tried doing this and this is where I've gone so far: $$\lim ...
3
votes
1answer
21 views

What is the value of $a$ so that this condition holds?

Let $f(x) \colon= x-x^2$, $g(x) \colon= ax$. Determine the value of $a$ so that the region above the graph of $g$ and below the graph of $f$ has area equal to $9/2$. Here $f(x) - g(x) = (1-a)x - x^2 ...
1
vote
2answers
55 views

How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx $$ in terms of the integral $$ \int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
1
vote
3answers
109 views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

How to evaluate the following integral $$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
8
votes
2answers
151 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
1
vote
1answer
58 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
9
votes
0answers
117 views
2
votes
1answer
42 views

Computing double integral

Find $$\iint\limits_D \sqrt{(x-10)^2+y^2}\hspace{1mm}dA$$ where $\{(x, y)\in D \mid x^2+y^2\leq 10^2\}$. I am not sure how to start, every way I have tried so far, ends up into something ugly. All ...
1
vote
3answers
111 views

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

How do we evaluate the integral $$I=\displaystyle\int_{\mathbb{R}} \dfrac{x\sin {(\pi x)}}{(1+x^2)^2}$$ I have wasted so much time on this integral, tried many substitutions $(x^2=t, \ \pi x^2=t)$. ...
1
vote
3answers
87 views

Evaluate $\int_{1}^{e}\frac{u}{u^3+2u^2-1}du.$

I'm trying to solve $$\int_{1}^{e}\frac{u}{u^3+2u^2-1}du.$$ My first approach was to factorise and then do a partial integration. However the factorisation ...
6
votes
0answers
105 views

An incorrect answer for an integral

As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed). $$ ...
3
votes
2answers
237 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
0
votes
2answers
61 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
0
votes
2answers
40 views

What is the area bounded by these curves?

Let $f(x) \colon = x^2$, $g(x) \colon= x+1$. Then what is the area bounded by the graphs of $f$ and $g$ between the vertical lines $x= -1$ and $x= (1+\sqrt{5})/2$? My effort: Since $$ f(x) - g(x) ...
10
votes
1answer
177 views
+100

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

I discovered the following conjecture numerically, but have not been able to prove it yet: $$_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47.\tag1$$ The equality holds ...
8
votes
2answers
155 views

An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

My brother asked me to calculate the following integral before we had dinner and I have been working to calculate it since then ($\pm\, 4$ hours). He said, it has a beautiful closed form but I doubt ...
8
votes
1answer
140 views

Integral: $\int_0^{\pi} \frac{x}{x^2+\ln^2(2\sin x)}\,dx$

I am trying to solve the following by elementary methods: $$\int_0^{\pi} \frac{x}{x^2+\ln^2(2\sin x)}\,dx$$ I wrote the integral as: $$\Re\int_0^{\pi} \frac{dx}{x-i\ln(2\sin x)}$$ But I don't find ...
4
votes
3answers
130 views

Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

I have heard that: $$\int_0^1\frac{x^a-x^{-a}}{x-1}dx=\frac1 a-\pi\cot(\pi a)$$ when $-1<a<1$. How would I prove this? That doesn't have an elementary indefinite integral, but the definite ...
9
votes
2answers
311 views

Evaluating $\int_0^\pi\arctan\left(\frac{\ln\sin x}{x}\right)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$ \int_{0}^{\pi} \arctan\left(\ln\left(\sin x \right) \over x\right)\,{\rm d}x $$ Mathematica and ...
1
vote
1answer
44 views

Area of solid revolution using integration.

When we calculate the volume of a solid generated by rotating a curve around $x$-axis, We divide it into disks. So ,we get $dv = \pi r^2 dx$. where $r=y$ and then we integrate. That OK, but when ...
18
votes
1answer
239 views

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

I discovered the following conjecture by evaluating the integral numerically and then using some inverse symbolic calculation methods to find a possible closed form: $$\int_0^\infty\frac{\ln ...
3
votes
0answers
94 views

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ dx$

Please help me to evaluate this integral: $$I={\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ dx.\tag1$$ Mathematica could not evaluate it in a closed form. A numerical ...
8
votes
1answer
155 views

Evaluate $\int_{0}^{\large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\frac{\ln{(\sin x)}}{\cot x}+\frac{\ln {(\cos x)}}{\tan x}\right)dx$

How do I find the value of this integral? $$I=\int_{0}^{\Large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\dfrac{\ln{(\sin x)}}{\cot x}+\dfrac{\ln {(\cos x)}}{\tan x}\right)dx$$ I tried ...
7
votes
1answer
194 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
1
vote
1answer
71 views

An integral representation for $\psi$

Let $\displaystyle \gamma$ denote the Euler constant defined by $\displaystyle \gamma := \lim\limits_{n \to \infty} \left(\frac11+\frac12+\cdots+\frac1n- \log n\right)$. Here is an integral for ...
0
votes
2answers
21 views

Determining the best way to compute a double integral

The question is: When graphed, this is what it looks like: I thought that the best way to do it would be with respect to y first, then x. The bounds: x/sqrt3 < y < sqrt(4-x^2) 1 < x ...
12
votes
0answers
139 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
4
votes
2answers
134 views
+100

Convergence of ${\large\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx$

Consider $$I(a)={\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx,$$ where $J_0(z)$ is the Bessel Function of the $1^{st}$ kind and $a>0$. Does this integral converge for any values of $a$? If so, is ...
3
votes
2answers
132 views

Evaluating $\int^b_a \frac{dx}{x}$ from the definition of the integral

I know that $$\int^b_a \frac{dx}{x}=\ln b-\ln a$$ I'm trying to evaluate this integral using the same method used in this answer: http://math.stackexchange.com/a/873507/42912 My attempt $\int^b_a ...
6
votes
2answers
191 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
4
votes
3answers
176 views

A closed form of $\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$$ I've tried substitution $y=\frac{1}{x}$ and $e^y=\frac{1}{x}$, but those didn't ...
0
votes
1answer
39 views

Continuity of $K(x,y)$ satisfying $g(x)= \int_0^1 \! K(x,y) f(y)\ \mathrm{d}x $ and $ \frac{d^3g}{dx^3} = f$

$g(x)$ is defined as the following : $$g(x)= \int_0^1 \! K(x,y) f(y)\ \mathrm{d}x $$ where $K(x,y)$ is continuous in $ 0 \leq x \leq 1 $ , $ 0 \leq y \leq 1 $, and $f(x)$ is continuous in $ 0 \leq x ...
8
votes
2answers
216 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
5
votes
1answer
126 views

Evaluation of $\int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$

Evaluation of $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$ $\bf{My\; Try:}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx\tag{1}$ ...
3
votes
3answers
103 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
2
votes
4answers
124 views

Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
1
vote
2answers
125 views

Evaluate $\int_0^1 \sqrt{2x-1} - \sqrt{x}$ $dx$

I'm trying to calculate the area between the curves $y = \sqrt{x}$ and $y= \sqrt{2x-1}$ Here's the graph: I've already tried calculating the area with respect to $y$, i.e. $\int_0^1 ...
7
votes
2answers
223 views

Proving that $\int_0^1\frac{x \log^2(1-x)}{1+x^2} \ dx = \frac{35}{32}\zeta(3)+\frac{1}{24}\log^3(2) -\frac{5}{96} \pi^2 \log(2)$

Could we possibly prove this result without using the polylogarithm? I know how to do it by polylogarithm means, but I want a different way. Is that possible? $$\int_0^1\frac{x \log^2(1-x)}{1+x^2} ...
2
votes
4answers
115 views

How to prove $\int_0^\pi \frac{dx}{2+2\sin x+\cos x}=\log3$?

How can we prove that: $$\int_0^\pi \frac{dx}{2+2\sin x+\cos x}=\log3$$ I don't have any ideas, the $f(\pi-x)$ thing doesn't work as well. Please help :)
2
votes
2answers
51 views

Examples of interesting integrable functions with at least 2 fixed points and an explicit inverse

What are some interesting functions I can use to demonstrate this integration trick: $$\int_a^b [f(x)+f^{-1}(x)]=b^2-a^2$$ I would like to know of some interesting functions where this trick is not ...
6
votes
1answer
70 views

Integral involving square root of sine and cosine

Is there any closed formula for $$ \int_{0}^{\pi/2} \dfrac{e^{-x}\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}} $$ I know $$ \int_{0}^{\pi/2} \dfrac{\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin ...