6
votes
2answers
92 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
99 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 ...
0
votes
0answers
51 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
4
votes
0answers
71 views

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...
0
votes
2answers
38 views

Volume of trig function around y-axis

I have this question and it's the first kind of question I'm doing involving finding volume so I just would like some help solving this question: Find the volume created by revolving the curve $ \ ...
4
votes
3answers
111 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 ...
11
votes
0answers
129 views

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

I found the following integral as a by product of another one. It has a nice closed form. $$ \displaystyle \int_{0}^{\pi} \arctan \left( \frac{\ln \sin x}{x} \right) \mathrm{d}x $$ ...
1
vote
1answer
40 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 ...
0
votes
0answers
15 views

Volume of Solid of Revolution

This problem is giving me some trouble: The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. x = (y − 5)^2, x = 4; ...
2
votes
2answers
46 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
0
votes
1answer
67 views

Seemingly hard integrals which are made easy via differentiation under the integral sign a.k.a Feynman Integration [on hold]

I recently discovered Differentiation under the integral sign a.k.a Feynman Integration and I read an article which says it can be substituted for contour integration. Therefore, I am assuming this ...
0
votes
5answers
75 views

Proving integrals of $f(\sin(x))$ and $f(\cos(x))$ are equal

Prove that for every continuous function $f$, $\displaystyle\int_{0}^{\pi/2} f(\sin(x))dx = \int_{0}^{\pi/2} f(\cos(x))dx $. I am not really sure how to tackle this, but nevertheless here is my ...
1
vote
4answers
89 views

Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
3
votes
1answer
44 views

Double integral proof, where is my mistake?

The bounds are 0 < x < b , 0 < y < b. $$ \int_0^b \int_0^b e^{-(x^{2}+y^{2})} dxdy $$ Since it is a square, x=y so we can write: = $$ (\int_0^b e^{-(x^{2}+x^{2})} )^{2} dxdy $$ = $$ ...
3
votes
3answers
259 views

Double integral proofs

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
3
votes
0answers
85 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 ...
0
votes
1answer
56 views

Just calculus, the integral = 0 and the argument inside integral = 0?

It is really hard for me to make a title to describe my question. Below is my question: Suppose $f(y-x)$ is a known Gaussian function defined as $$ f(y-x) = \frac{1}{\sqrt{2\pi}} \exp ...
8
votes
1answer
141 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 ...
1
vote
0answers
14 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
6
votes
1answer
148 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
68 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
1answer
20 views

Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
2
votes
1answer
65 views

Evaluate the area of the region bounded by the ellipse, where is my mistake?

$ (10x^2+6xy+y^2=2)$ => $ ((x/\sqrt2)^{2} + ((3x+y)/\sqrt2))^{2} = 1 $ so if I change the variables to $u$ and $v$, $u = x/\sqrt2$ $v= (3x+y)/\sqrt2) $ Then my bounds of integration become $-1 ...
0
votes
0answers
16 views

How do you obtain the version of Simpson's rule required as well as deduce the composite integration rule? [on hold]

Consider the function $$g(x)=f(a+(x−1)h)$$ and obtain a version of Simpson’s rule applicable to an integral $$\int_{a+h}^{a−h}f(x)dx.$$ Then deduce the composite integration rule ...
2
votes
2answers
91 views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}dx$

Evaluation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx ...
0
votes
1answer
34 views

How do you solve the second part of the question where i am required to derive Simpson’s integration rule?

When $v(x) = A + Bx + Cx(x − 1)$ show that $$\int_0^2v(x)dx= 2A + 2B + \frac23.$$ By choosing A,B and C so that $y = v(x)$ fits a given curve $y = g(x)$ at $x = 0$, $x = 1$ and $x = 2$ derive ...
1
vote
1answer
39 views

What function to use to get geometric mean in trapezoidal rule?

When deriving a trapezoidal rule an integral of $f(x)$ is switched to integral of new function $g(x)$ approximating the first one $$\int_a^b {f(x)dx}\approx \int_a^b {g(x)dx}$$ where $g(x)$ is a ...
4
votes
1answer
114 views

How can I evaluate this indefinite integral? $\int\frac{dx}{1+x^8}$

How do I find $\displaystyle\int\dfrac{dx}{1+x^8}$? My friend asked me to find $\displaystyle\int\dfrac{dx}{1+x^{2n}}$ for a positive integer $n$. But looking up I am getting pretty noisy answer for ...
1
vote
1answer
22 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
-1
votes
3answers
66 views

Find $\frac{dG}{dx}$ of $G(x)=\int_0^{x^2}\frac{dt}{t^2+4}.$ [on hold]

Define $$G(x)=\int_0^{x^2}\frac{dt}{t^2+4}.$$ What is $\displaystyle\frac{dG}{dx}$? How do I approach this question? What are the steps? What is the solution?
3
votes
1answer
45 views

Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
1
vote
1answer
43 views

Existence of a function with certain integral properties

Does there exist a non-negative Borel-measurable function $g:\mathbb [1,\infty)\to[0,\infty)$ such that \begin{align*} \int_1^{\infty}g(y)^2\,\mathrm dy<&\,\infty,\\ ...
0
votes
2answers
20 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 ...
0
votes
3answers
21 views

Definite integral fractional exponent in the denominator

I have come across this question and I cannot understand the step highlighted. I would have expected that the fractional exponents of the terms in the numerator would have a negative value after ...
3
votes
2answers
41 views

Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
12
votes
0answers
122 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 ...
2
votes
3answers
38 views

Integration Trig Substitution

After making the correct trig substitution what does the integral of $\dfrac{1}{\sqrt{9-x^2}} dx$ reduce to without solving the equation? I reduced it down to the integral of ...
1
vote
1answer
96 views

Evaluation of $ \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$ [duplicate]

Evaluation of $\displaystyle \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$ $\bf{My\; Try::}$ Given $\displaystyle \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx = \int ...
3
votes
2answers
69 views

Without Lebesgue

Everyone knows following problem. Let $f$ be positive function on $[0,1]$ and there exist $I = \int_{0}^{1}f(x)dx$. Prove that $I>0$. (recall that there are only two cases: $I=0$ or $I>0$. NOT ...
3
votes
2answers
129 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
187 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 ...
1
vote
1answer
29 views

variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], ...
1
vote
1answer
146 views

How do I find the equation from this differential equation?

I have this thing in a video game that I'd like to optimize using math instead of trying random combinations. In every game loop, the game calculates the new value ("newRotorEnergy" in the equations ...
3
votes
3answers
163 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 ...
2
votes
2answers
55 views

Integrating 1/x

The standard definition of integrating $\frac{1}{x}$ is: $$ \int \frac{dx}{ax + b} = \frac {1}{a} \ln |ax + b| + K $$ Now, if I'm understanding the "constant factor rule", that is: $$ \int k ...
8
votes
2answers
209 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 ...
3
votes
3answers
98 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 ...
1
vote
2answers
55 views

Finding Cauchy principal value for: $ \int_1^\infty \frac{ a x^2 + c }{x^4 - b x^2 - c} \mathrm{d}x $

I need to solve the integral $ \displaystyle \mathcal{P} \int_1^\infty \frac{ a x^2 + c }{x^4 - b x^2 - c} \mathrm{d}x $, where $\mathcal{P}$ is the Cauchy principal value, $ - 1 \leq c \leq 1$ and ...
2
votes
2answers
142 views

integrate $ \int \frac {x dx}{\sqrt {1+x^{10}} } $

This is a tough one. Thanks. $$\int \frac {x dx}{\sqrt {1+x^{10}} } $$ This is not a homework problem. I spend 10 hours over the course of 3 days on this. I tried: 1) substituting u for x^5 to get ...
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 ...