Questions about the evaluation of specific definite integrals.

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Closed form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} ...
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1answer
35 views

$\int_0^{2 \pi} \cos(x)e^{i (a \cos(x) + b \cos^2(x)} dx$ and $\int_0^{2 \pi} \cos^2(x)e^{i (a \cos(x) + b \cos^2(x)} dx$

I am currently dealing with the two integrals in the title and I want to find out, when their real part of their imaginary part vanishes ( so for which constellation of $(a,b) \in \mathbb{R}^2 ...
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8answers
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Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
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1answer
40 views

Integral $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$

I'm looking to calculate $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$. Mathematica10 can't integrate this, but numerical integration gives an answer of ...
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0answers
44 views

Calculation of this integral

Evaluating $$\int_0^1\frac{\log^2(x)\log(1+x^2)}{1-x^2}dx$$ I found $- \dfrac{\pi^4}{32}+2G^2+\dfrac74 ζ(3)\log2 $ where G is the Catalan's constant.
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1answer
25 views

Calculating area relative to the y-axis

I was asked to calculate the area of the region bounded by the following graph: $$ y = x^2+4x ; y=0$$ I substituted $y$ in order to get $x = 0$ 0r $x=4$. Now I would like a little bit of help to get ...
5
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2answers
69 views

Calculating value of $\pi$ independently using integrals.

Recently I noticed this integral: $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}7-\pi\approx0$$ Which is a very interesting result which gives us the value of ...
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7answers
102 views

How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$

How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$) $$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$ Integration by parts doesn't seem to make ...
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3answers
82 views

Integral $\int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x}$

Integrate: $$ \int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x} $$
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0answers
60 views

Is there a formal proof of this basic integral property?

This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth ...
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1answer
30 views

Use the shell method to find the volume of the solid obtained by rotating the region bounded by the given curves

$$y=\sqrt{x}$$ $$y=0$$ $$x=1$$ about x=-1 Does this set up look alright: $$V = \int_0^1 2 \pi (x+1)(\sqrt{x}) dx$$
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1answer
42 views

How to find $F(x) = \int_x^{x^2} (2+\sqrt t )\, dt$ ?

I have this problem: $$ F(x) = \int_x^{x^2} (2+\sqrt t )\, dt $$ I have to solve the integral. I got $2x^2+\frac{2x^3}{3}-2x-\frac{2x^{3/2}}{3}$ However, I don't think that it correct.
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4answers
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Calculus find extreme values of integral

I have this problem: Ineach of Exercises 48-51, a definite integral is given. Do not attempt to calculate its value $V$. Instead, find the extreme values of the integrand on the interval of ...
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3answers
64 views

Evaluate the integral $\int_0^{1/4}\frac{x-1}{\sqrt{x}-1}\mathrm dx$

so I have this Integral I have to solve without a calculator. $$\int_0^{1/4}\dfrac{x-1}{\sqrt{x}-1}\mathrm dx.$$ How would I go about finding the antiderivative of that fraction?
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1answer
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Evaluating an integral with unspecified functions $f,g$, given other integrals with these functions

Suppose that $$\int_6^8(3f(x)-x)\,\mathrm dx=6$$ and $$\int_8^6(2x+4g(x))\,\mathrm dx=-8$$ Evaluate $$\int_8^6 (f(x)-5g(x))\,\mathrm dx$$ I have a problem. So, this one question asks me ...
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0answers
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Prove the given two integrals are not equal

I am stuck with following problem: Prove the following two integrals are not equal: $$ \int_{-\infty}^{\infty} p(y-c)\log \big(p(y-c)+p(y+c)\big)dy \neq \int_{-\infty}^{\infty} p(y+c)\log ...
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1answer
30 views

Proving an integration with a modified Bessel function and an exponential

I am trying to prove the following identity: where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of the ...
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3answers
125 views

Help me with this definite integral

I don't know how to solve this definite integral, maybe the solution is evident but i don't see it : $\int_0^\frac{\pi}{2} \frac{\cos^3(x)}{(\cos(2x) + \sin(x))}\,dx$
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1answer
174 views

determine if a function is periodic

Let $f$ be a continuous and integrable function on $[a,b]$ such that $$\int_a^b f(x)\,\mathrm{d}x = 2$$ and for every $t_1,t_2$ such that $\displaystyle t_2 -t_1 = \frac{b-a}2$ ...
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0answers
108 views

Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$$ My first thought was to use beta function, but it's hard to get ...
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Integral formulation for LDE

I am trying to put the system in a integral formulation. All goes well for the first integration as I obtain What I don't know is how to perform the second integration in this last term. My ...
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1answer
51 views

Another parametric integral relating to hyperbolic function

if $0<a\leq1$, then canwe get a closed form of $$I(a)=\int_0^\infty\frac{x}{\tanh x}\frac{1}{\cosh^2(ax)}dx.$$ In fact,if $a=1$,$I(a=1)=\pi^2/8$.
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1answer
52 views

Integral with quadratic square root inside trigonometric functions

Is there anyway to solve $\displaystyle \int t \frac{\sin \left(\frac{t}{2} \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}\right) }{ \sqrt{ a \left(t+ ...
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1answer
16 views

Advanced convolution u-substitution involving error functions

In finishing the evaluation of a partial differential equation, I've arrived at a stage of a convultion integral I'm stuck at. I have to evaluate the following integral ...
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2answers
22 views

Definite integral of trig function

I'm looking for some assistance on the following problem: Let $$ T(x) = \int_{4r^3}^{4} tsin(t^3)dt $$ Find $$T'(r)$$ I'm struggling to find the antiderivative of the sine function, particularly as ...
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2answers
144 views
+100

Closed form of $I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
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1answer
30 views

Integral of a case function

Assume the following function, with $x, a, b \in \! \mathbb{R}$ $$ f(x,a,b) = \begin{cases} x+a+b & \mbox{for } ~ a-b \le x \le a+b \\ 0 & \mbox{elsewhere} \end{cases} $$ How can ...
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0answers
26 views

Integral analysis

So lately I had to work out mathematically the predicted relation between radiation intensity (I) and thickness of the obstacle (A). However, I found on the internet some notations using integrals ...
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0answers
104 views
+200

Closed form for integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ From Vladimir ...
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Looking for advice with the following integral

I have the following integral to evaluate: $$\frac{1}{f(t)} \int_0^t s^m f(s) \sin(ps) \mathrm{d}s \quad m,p \in \mathbb{R}$$ I'm unable to proceed with this integral as it is non-trivial. Even using ...
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1answer
65 views

Numerical value of $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

Could somebody give me a numerical value for this integral? $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$
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Expected Value: Approximation using integrals and observed data (of unknown function)

For simplicity, say my dataset assumes y=x^2 (but is not intended just for this dataset, it is for observing game data outputs and trying to figure out a way to approximate their distribution). ...
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1answer
12 views

error order evaluation in taylor expansion of a definite integral

I have a function $g(x)=f(x)e^{-x}$ and i want to consider the following integral: $\int_{0}^{\infty}g(x)dx$. Since $f(x)$ is a complicated, but monotonic decreasing, function in the interval ...
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0answers
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Calculus Single Variable: Find max and min of hard to graph function

Consider the function F defined by F(x)= integral from 0 to x of $t|sint(t)|dt$. Find the absolute maximum value and absolute minimum value of y=f(x). I know there's one at x= zero but the ones ...
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1answer
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+200

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
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1answer
22 views

order of integrals with independent limits

I was wondering if the following is true assuming that the limits are independent (like constants) $$ \int_{\alpha}^{\beta} \int_{\gamma}^{\psi} {xy} dx dy = \int_{\gamma}^{\psi} ...
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1answer
14 views

Error estimate of definite integral of a taylor expanded function

If I consider a monotonic decreasing function $f(x)$ in the interval $[0,+\infty[$, and I consider the definite integral $\int_{0}^{+\infty}f(x)\,\mathrm{d}x$. What is the error committed if I compute ...
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1answer
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Motivation behind parameters

This article shows a technique of evaluating a definite integral by introducing a suitable parameter. This however doesn't throw light on motivation for introducing that particular parameter. For ...
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Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
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0answers
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Find $\int \tan(\tan x)\hspace{1mm}dx$

Find $\int \tan(\tan x)\hspace{1mm}dx$ This is an Interesting problem, which I have been trying from different directions, nothing seems to work, its been a day on this one. Can anyone figure out ...
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1answer
37 views

Finding area between two cosine curves

I must to find the area between these two curves: $$y = 2 \cos 7x, y = 2 − 2 \cos 7x$$ $$0 ≤ x ≤ π/7$$ And this is all I have so far: $$ 2\cos7x=2-2\cos7x $$ $$4\cos7x=2$$ $$\cos7x=1/2$$
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1answer
380 views

Closed form for ${\large\int}_0^1\frac{\ln^3x}{\sqrt{x^2-x+1}}dx$

This is a follow-up to my earlier question Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$. Is there a closed form for this integral? ...
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1answer
45 views

How to simplify this complex integral? [closed]

How to approximate this integral as a function of a and b? $$\int_0^\pi\int_0^{2\pi}\sqrt{(a-b\sin\varphi\cos\theta)^2+(b\cos\varphi)^2+(b\sin\varphi\sin\theta)^2}d\theta d\varphi$$ where a and b ...
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4answers
229 views

Finding the definite integral of a function that contains an absolute value

The integral in question is this: $\int_{-2\pi}^{2\pi}xe^{-|x|}$ My attempt: Since there is a modulus, we split it up into cases. I'm not really sure which cases to split it into, do I just ...
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0answers
31 views

Could someone help find the shell height?

I am trying to solve this problem and have been going at it for 3 hours and not getting anywhere. I think I am suppose to have everything in terms of y but the x equals functions are throwing me off. ...
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2answers
33 views

Find $\int t\sin^{-1}t\hspace{1mm}dt$

Find $\int t\sin^{-1}t\hspace{1mm}dt$ How do we approach this question, is there a simple way to integrate
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1answer
39 views

Integral from 0 to 16 of $\sqrt{x}/(x-4)$?

$$\int_{0}^{16}\frac{\sqrt{x}}{x-4}dx$$ So I'm letting $u=\sqrt{x}$, $du=1/2\sqrt{x}$, $u^2=x$ and $dx=2\sqrt{x}du$. I just don't really know what to do from here. I am trying different things and ...
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3answers
43 views

Using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$?

So using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$, and we're given $\int_1^\infty\frac{1}{4x^2}dx$. So I have been trying to set up an inequality to use, but I can't seem ...
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2answers
18 views

Calculate the Area of the space defined by two lines $\varepsilon_{1},\varepsilon_{2}$ and a curve $c_{1}$

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
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4answers
84 views

Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...