Questions about the evaluation of specific definite integrals.

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43 views

How Prove this integral is diverge $\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$

Show that this following integral is divergent (or diverges, if you prefer) $$\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$$ I know when $x=0,1$ are singularities of the function and I want use this ...
0
votes
0answers
10 views

Error terms for Composite Newton-Cotes formulas

There are four general error formulas for Newton-Cotes rules (according to closed/open formulas and odd/even points). My question is: are there four general error formulas for composite Newton-Cotes ...
3
votes
1answer
53 views

Taylor series of a definite integral

Consider the function of a parameter $\alpha > 0$, given by $$f(\alpha) = \frac{2}{\sqrt 2\pi} \int_0^\infty e^{\dfrac{-x^2}{2\alpha^2}}\cosh(x)\log\cosh(x) dx.$$ From Wolfram-alpha, it seems that ...
2
votes
1answer
44 views

Prove or disprove following integral.

Assume $L$ a constant, and assume $x$ real. Is the following equation true? $$ \int_{-\infty}^\infty\frac{1}{k^2}\exp(-ikx)dk = \frac{L}{|x|} $$ If it is true, find the value of $L$. If it is not, ...
1
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2answers
60 views

Find $\int_0^2 \arctan(\pi x)-\arctan(x)\, \mathrm dx$

Find $$\int_0^2 \arctan(\pi x)-\arctan(x)\, \mathrm dx$$ The hint is also given : Re-write the Integrand as an Integral I think we have to Re-write this single integral as a double integral and ...
7
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0answers
40 views

Hard sum with harmonics numbers

Prove or disprove that $S=\displaystyle\sum_{n=1}^{\infty}\frac{{H_n^{2}}~{H_n^{(2)}}+3{H_n^{(4)}}}{n~2^n}=\frac{25}{16}\zeta(5)+\frac{7}{8}\zeta(2)\zeta(3)$.
1
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1answer
58 views

$ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $?

I have the following implication, and I need to determine whether it's true: $ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $ I tried solving ...
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0answers
37 views

$F(x) = \int_0^x f(t)~dt \implies F(1)=f(0)+\int_0^1(1-t)f'(t)~dt$?

f is differentiable and has a continuous derviative, and $F(x) = \int_0^x f(t)~dt$. Based on this assumption, I have the following statement which I need to determine whether it's true or false: ...
1
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2answers
50 views

f is even or odd, prove that f^2 is even

I need to verify whether a statement is correct or false. The statement is as following: If the function f is either odd or even, then the function f^2 is even. To my understanding, the statement is ...
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0answers
18 views

Integrals: Average(f)*Average(g)=Average(f*g) [on hold]

So I've got everything but question #3 here. I understand that it isn't simply (1/4)(1/4)=16. And also not (1/4)(1/4)(1/4)=1/64. But I can't think of what else it might be. It isn't discussed in the ...
0
votes
2answers
52 views

Evaluate $\displaystyle\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$ using residue method [on hold]

This is a real integral but I want to evaluate it using residue integration method $$\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$$
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1answer
32 views

Expressing limit of sum definite integral

Evaluate limit by expressing it as a definite integral. ...
0
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1answer
32 views

Problem with this question on solid of revolution

Calculate the volume of a revolution solid obtained by rotation around the x-axis, the region bounded by the graph of $y=e^x$, $-1\le x \le1$ and the x-axis. Thanks in advance, and sorry about my ...
2
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2answers
86 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculation of $\displaystyle \int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$ given that $ a>b>0$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{\sin^2 ...
0
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2answers
105 views

Evaluating $\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} dy$

I was trying to find arc-length of $x = \sqrt{y}-7y$ So basically right now I am stuck with this $$\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} \,\mathrm dy$$ $$\int^{4}_{1} ...
6
votes
2answers
61 views

Integration $\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R$.

Give a compact form for the solution of integral: $$\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R,k\in\mathbb N$$ any suggestions please?
4
votes
6answers
154 views

Show that $\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$

$$\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$$ My intuition telling me there might be an $\arctan$ coming up, but I don't know how to do this ...
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2answers
37 views

Convolution Integral and dummy variable

I have no idea how to find a dummy variable or start with this problem. I'm given the function: $$ {\rm g}\left(\, x\,\right) =\int^{x}_{0}\left(\, x - t\,\right){\rm f}\left(\,t\,\right)\,{\rm d}t ...
2
votes
3answers
48 views

Integration involving square root and negative power of $x$

$$ f(x) = \int\sqrt{1+x^{-2/3}}\,\mathrm dx $$ What I have attempted: $$ \int\left(\frac{\sqrt{x^{2/3}}}{\sqrt{x^{2/3}}}\right)\sqrt{1+x^{-2/3}}\,\mathrm dx ...
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0answers
14 views

Find the volume of the 3D solid.

Let $\delta$ be the region bounded by the graphs of the functions $f(x) = x^2$ and $g(x) = 2 x^2$. Find the volume of the solid generated by revolving $\delta$ around the line $x = 1$.
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2answers
82 views

Determine if $\displaystyle \int_3^{\infty}\frac{x+1}{\sqrt{x^4-x}}\,dx $ converges

Determine whether the following integral is convergent or divergent without evaluating it. (Whichever answer is correct, you must show why it is true.) $$ ...
4
votes
1answer
68 views

Integral of cosine over a quadratic

I need help with the following integral: $$ \int_{-\pi}^{\pi}{\cos\left(\, ax\,\right) \over 1-bx^{2}}\,{\rm d}x $$ The constants $a$ and $b$ are both real and positive. Any help will be ...
11
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6answers
244 views

Ways to prove $\displaystyle \int_0^\pi dx \dfrac{\sin^2(n x)}{\sin^2 x} = n\pi$

In how many ways can we prove the following theorem? $$I(n):= \int_0^\pi dx \frac{\sin^2(n x)}{\sin^2 x} = n\pi$$ Here $n$ is a nonnegative integer. The proof I found is by considering ...
3
votes
2answers
53 views

Improper Integral of $\int\frac{dx}{(2x-1)^3}$

Improper Integral of $$\int_{-\infty}^0\frac{dx}{(2x-1)^3}$$ from Anton Calculus 8th Edition, page 576, question 9. Answer is $-\frac{1}{4}$ but I'm finding $-1$ The integral, substituting ...
1
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3answers
88 views

Evaluating $\displaystyle\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$

Evaluate the following limit: $$\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$$ Any advice on how to tackle this problem ?
6
votes
3answers
80 views

Why is the area under $\frac1{\sqrt{x}}$ finite and the area under $\frac1x$ infinite?

If this integral is calculated the normal way, $$\int_0^1 \frac1{\sqrt{x}} dx = 2\sqrt{1}-2\sqrt{0}=2$$ However, the graph of $\dfrac1{\sqrt{x}}\to \infty$ as $x\to 0$, so the area under the graph ...
2
votes
1answer
39 views

Using $\log$ and $\ln$ in Integration [duplicate]

I found in some integral equations where they use $\log(n)$ and in some other with $\ln(n)$. Suppose $$ \int_{n_0}^{\large\frac{n_0}{2}} \frac{1}{n}dn $$ Which formula should I use ? $$ \log(n)\ ...
0
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1answer
32 views

shell method not giving right answer

The soot produced by a garbage incinerator spreads out in a circular pattern. The depth, $H (r)$, in millimeters, of the soot deposited each month at a distance $r$ kilometers from the incinerator is ...
4
votes
2answers
75 views

how to compute this definite integral if possible?

how to solve this integral? $$\int_0^a\int_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac{3}{2}}$$ my attempt $$ \int_0^a\int_0^a\frac{dx \, dy}{(x^2+y^2+a^2)^\frac{3}{2}}= ...
1
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1answer
29 views

Convergence of a simple Integral

Let $f$ be continuous on the interval $[0, 1]$. How can I show that $$\lim_{n\to\infty}\int_0^1 f(x)\sin nx\,dx = 0 ?$$
0
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1answer
40 views

Proving a Simple Integral with Exponents

Let $f$ be differentiable in $[a,b]$. How can I show that $$\exp\left(\frac{1}{b-a} \int_a^b f(x)dx \right) \le \left(\frac{1}{b-a}\right) \int_a^b \exp(f(x)) dx $$
7
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2answers
66 views

Evaluating $\int_0^{\pi/3}\cosh^2\left(x/\sqrt{2}\right)\tan^3x \:dx$

I've been told that this integral admits a closed form $$ \int_0^{\Large\pi/3}\cosh^2\left(x/\sqrt{2}\right)\tan^3x \:dx$$ But an integration by parts with $u'(x)=\cosh^2\left(x/\sqrt{2}\right)$ and ...
10
votes
4answers
276 views

How to evaluate $\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$

Evaluate $$ \int_{0}^{1} \arctan^{2}\left(\, x\,\right) \ln\left(\, 1 + x^{2} \over 2x^{2}\,\right)\,{\rm d}x $$ I substituted $x \equiv \tan\left(\,\theta\,\right)$ and got $$ ...
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votes
2answers
67 views

Find this integral $\int_{0}^{1}f(x)dx$ [closed]

let the function $$f(x)=\begin{cases} 1&x\in\{1,\dfrac{1}{2},\dfrac{1}{3}\cdots,\dfrac{1}{n},\cdots\}\\ 2&x\in other \end{cases}$$ Find this integral $$\int_{0}^{1}f(x)dx$$ where this ...
7
votes
0answers
118 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
4
votes
1answer
64 views

Evaluation of $\int_{-v/2}^{v/2} \sqrt{1-\left(\frac{t}{7}-\frac{v}{14}\right)^2} \sqrt{1-\left(\frac{t}{7}+\frac{v}{14}\right)^2} dt$

I need to get the value of the following definite integral $v\in \mathbb R^+$ $$\int_{-v/2}^{v/2} \sqrt{1-\left(\frac{t}{7}-\frac{v}{14}\right)^2} \sqrt{1-\left(\frac{t}{7}+\frac{v}{14}\right)^2} ...
0
votes
1answer
43 views

Table of Integrals. Which one to use?

I just finished my exam and there was this question. It asks us for which form in the table of integrals to use. $\int(x-3)\sqrt(6+6x-x^2)$ I did completing the square and got into this form ...
1
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1answer
36 views

$\int_{-\pi}^{\pi}\sinh(x) \sin(nx)dx$, using complex transformation.

$$\int_{-\pi}^{\pi}\sinh(x) \sin(nx)dx$$ This integral can be done by using integrate by parts twice. However, I was thinking if this can be done by using some complex transformation (I haven't done ...
2
votes
1answer
47 views

using contour integrals

Let $ \gamma (t)= e^{it} $ where $0 \leq t \leq 2 \pi.$ Evaluate $\int_{\gamma}$ $e^{z}$ $dz$ . Use the result to show that $\int_{0}^{2\pi} e^{\cos(t)}\cos(t+ \sin(t)) dt = 0$. I have worked out ...
2
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0answers
37 views

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
0
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1answer
24 views

Strict upper and lower bounds of a sum (Big-Theta)

I am trying to find a function f(k) such that $S_k=\sum_{n=1}^{k^2-1}(\lfloor\sqrt{n}\rfloor)=\Theta(f(k))$. What I have done so far: Ignoring the floor asymptotically we get: ...
7
votes
2answers
52 views

Find $\mathcal{L}\left\{\cos^3\left(t\right)\right\}$

I began by breaking the problem up as follows: \begin{align} \mathcal{L}\left\{\cos^3\left(t\right)\right\}=\int_0^\infty e^{-st}\cos^3\left(t\right)\:dt & = \int_0^\infty ...
0
votes
3answers
40 views

Proof with Fundamental Theorem of Calculus

If $f'(t)\leq 10$ for $0 \leq t \leq 5$ and $f(0)=3$: How I can explain with $\int_a^b f'(t)\,dt=f(b)-f(a)$ what the maximum of $f(5)$ is?
0
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2answers
31 views

Proof with integral properties

I'm trying to explain/looking for an answer whether a positive function $u=f(t)$ exists, for which $\int_{t=0}^{1}u\,dt = \int_{t=1}^{0}u\,dt$ is true. As we all know, the correct theorem is ...
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votes
2answers
67 views

Find $\int_0^{2\sqrt{\pi}}\int_{x/2}^{\sqrt{\pi}}\sin(y^2)dydx$ [on hold]

Find $$\int_0^{2\sqrt{\pi}}\int_{x/2}^{\sqrt{\pi}}\sin(y^2)dydx$$ Not sure how to start
14
votes
6answers
290 views

Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques

Is there a way to compute this integral, $$\int_0^1 \frac{x^k-1}{\ln x}dx =\ln({k+1})$$ without using the derivation under the integral sign nor transforming it to a double integral and then ...
0
votes
1answer
32 views

Integral from $e^{-itx}$ over $\mathbb R$

As a part of my task considering characteristic functions I have to compute $\int_{\mathbb R} e^{-itx}dx$ The result i get is $\frac{1}{-it}e^{-itx}|^{\infty}_{-\infty}$, but I don't really know ...
8
votes
6answers
175 views

Evaluate $\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx$ (solution verification)

I tried to find the integral $$I=\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx \tag1$$ I substituted $x=t^2, 2tdt=dx$ and chose $\log x$ and $\sqrt{x}$ to be principal values. We have ...
2
votes
2answers
79 views

Closed form for $\int_0^\infty e^{-x}\sin^a(x)dx$

Can we find a closed form for $$I(a)=\int_0^\infty e^{-x}\sin^a(x)dx$$ Mathematica can easily find closed form for integer $a$: \begin{align*} I(0)&=1\\ I(1)&=1/2\\ I(2)&=2/5\\ ...
3
votes
1answer
46 views

What am I doing with this triple integral?

I am new here and hope my question is clear and is straight to the point. The following is a form of an integral I am trying to compute. $$\int_{x}\int_{y}\int_{z} f(x,y,z) g(x,y)\ dz \ dy \ dx \ ...