Questions about the evaluation of specific definite integrals.

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5
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1answer
47 views

Evaluate $\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$

I want to find a quick way of evaluating $$\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$$ This problem appeared on the qualifying round of MIT's 2014 Integration Bee, which leads me to think ...
0
votes
1answer
53 views

Evaluate $\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$

My question is: Evaluate $$\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$$ for $k=0,1,...,(n-1)$ and $n \in \mathbb{N}$. I've tried integration by parts but without much success. Any ...
0
votes
2answers
18 views

How to find volume of the given solid analytically?

Here is the question - I am able to visualize the solid, but how do I find its volume? I'm unable to figure out the 2D structure that when rotated, produces this solid. Please help. Edit: The ...
2
votes
1answer
66 views

What is $\int_0^1 \frac{\log(x+1)}{x^2+1}dx$ [duplicate]

It's so deceptively simple and none of the usual techniques are working. Any and all insights are welcome.
1
vote
0answers
40 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...
0
votes
1answer
27 views

The Gherkin (an egg shaped building) - equation for the curve in order to calculate the surface area of revolution

I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that ...
1
vote
2answers
38 views

How we can prove that: $\sum _{_{k=n}}^{^n}\:f\left(\frac{k}{n}\right)\le n\cdot log\left(2\right)$?

$f:\left[0,1\right]\rightarrow R,\:f\left(x\right)=\frac{1}{1+x}$ and we have to show that $\sum _{_{k=n}}^{^n}\:f\left(\frac{k}{n}\right)\le n\cdot\log\left(2\right)$.What I know is just that: ...
1
vote
1answer
30 views

An integral related to the derivative of Legendre polynomials

I want to calculate the integral $$ I=\int_{-1}^{1} \Big(\frac{\mathrm{d}P_{n+1}(t)}{\mathrm{d}t}\Big) \Big(\frac{\mathrm{d}P_{m+1}(t)}{\mathrm{d}t}\Big) \mathrm{d}t $$ where $P_n(t)$ is Legendre ...
1
vote
1answer
60 views

Definite integral involving 2015

Evaluate $$\displaystyle\int_{2}^{2014} \frac{\log \left( 2015 - x\right )}{\log \left( 2015 - x\right ) + \log \left( x - 1\right )} \mathrm{d}x$$ I got the solution using software, and it is a ...
0
votes
0answers
45 views

how to solve this integral involving any square root

how to solve the integral $\int\sqrt{\alpha+\beta e^{\gamma t}}dt$ i got this integral from the problem Given that the velocity $v$ of a body $t$ segonds after passing a point $O$ is found by ...
0
votes
3answers
79 views

Integrating $f(x) = 1/x$ from $x=a$ to $x=\infty$

Can the integration of $f(x)=1/x$ from $x=a > 0 $ to $x=\infty$ ever be finite? That is, can $\int_{x=a}^{\infty} 1/x$ be finite?
2
votes
0answers
35 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = ...
4
votes
3answers
105 views

Compute $\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$

Given $$\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$$ I couldn't evaluate this integral. My only idea here was evaluating this as integration by parts. \begin{align} \int\frac{x ...
2
votes
1answer
52 views

Where am I wrong in the following problem?

We have: $f:R\rightarrow R,\:\:f\left(t\right)=At^2-2Bt+C,\:where\:A=\int _1^2\:\frac{1}{x^2}dx,\:B=\int _1^2\:\frac{e^x}{x}dx,\:C=\int _1^2\:e^{2x}dx$ and we need to show that ...
4
votes
2answers
91 views

Evaluate $\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$$ For this integral, I have tried using integration by parts and then ...
0
votes
1answer
26 views

Divergence of $\iint \text e^{-(\tau_1-\tau_2)}\,\theta(\tau_1-\tau_2)\,\text d ^2\tau$

Does this integral ($\alpha>0$) $$ I=\int_{-\infty}^\infty\text d \tau_1 \int_{-\infty}^\infty\text d \tau_2 \; \text e^{-\alpha(\tau_1-\tau_2)}\theta(\tau_1-\tau_2) $$ diverge? Here $\theta$ is ...
1
vote
2answers
32 views

Misunderstanding inequalities of integrals

We have to prove the following inequalities: 1) to show that $\frac{2x}{\pi }<sin\left(x\right)<x,\:and\:after\:1-e^{-\frac{\pi }{2}}\le \int _0^{\frac{\pi ...
2
votes
2answers
21 views

A question involving finding values in integrals?

Let p(x) be a continuous function such that $\int_2^3{p(x)}dx$=$c\cdot\int_0^2{p(\frac{x+4}{2}})dx$ then find the value of c? I am thinking of dividing the integral on the left hand side into two ...
0
votes
0answers
6 views

Hydrostatic Force on a submerged plane.

I am having trouble with question 5, it reads (ignore the Riemann sum part) : Here is what I did, and where did I go wrong?. The answer the book gives is: $6.7\cdot 10^4N$. Thank you.
1
vote
1answer
53 views

Evaluate $\int_{0}^{1} \frac {\ln x}{1-x^2} \mathrm{d}x $

I found this question in a reference book: $$\int_{0}^{1} \frac {\ln x}{1-x^2} \mathrm{d}x $$ Can Anyone give me Idea how do I begin solving this?
1
vote
2answers
25 views

sum approximation of a Lipschitz-continuous function

Let $f: [0, 1] \to \mathbb{R}$ be a Lipschitz continuous function with a Lipschitz constant $L > 0$, meaning: $$|f(x) - f(y)| ≤ L|x - y| \space\space\space \forall x, y \in [0, 1]$$ For the ...
0
votes
3answers
61 views

Integrate problem

We have to integrate $\int _0^{\pi }\:\left|\sin\left(2x\right)\right|dx$ and in my book, they split integral: $\int _0^{\pi }\:\left|\sin\left(2x\right)\right|dx=\int _0^{\frac{\pi ...
1
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0answers
39 views

Calculate an integral with delta function

In order to calculate the integral $$ f(x) = \frac{2}{\pi}\int_0^{\pi/2}\delta\Big(x-\sqrt{1\pm\sqrt{1-\beta^2\sin^2t}}\Big)\mathrm{d}t $$ where $\beta\in(0,1]$. I am hunting for a better solution, ...
1
vote
3answers
50 views

Integral of a tangent function

$$ \displaystyle {\int_{0}^{z}} \sqrt {1 + \tan^2(\dfrac{\pi}{4} \dfrac{z}{H} )} dz $$ _ $$ gives $$ _ $$ \dfrac{4H}{\pi} {\sinh^{-1}} ( {\tan \dfrac{\pi}{4} \dfrac{z}{H} } ) $$ Please advise ...
-4
votes
0answers
58 views

$\int_{1}^{\infty}\frac {P(x)}{2x+1}\,dx = \frac {-3}{4}+\frac {1}{4}\log 2 + \frac {1}{2}\log 3$ [on hold]

I am looking for the possible ways to solve it. $$\int_{1}^{\infty}\frac {P(x)}{2x+1}\,dx = \frac {-3}{4}+\frac {1}{4}\log 2 + \frac {1}{2}\log 3$$ Where $$P(x)=x-[x]-\frac {1}{2}$$ P.S. $[x]$ is ...
3
votes
1answer
47 views

How we can show that $\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{2n}\right)$

We have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}\:dx,$ and we need to show that$\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)$ I write ...
0
votes
2answers
84 views

Integration of $x^n e^{-x} dx$

I've been trying solve this, and even though I feel I'm really close to the answer- I'm quite unsure of the actual answer. The question is a definite integral $$\int_{0}^{\infty} \frac {x^n} ...
0
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0answers
6 views

Setting up triple integral in spherical coordinates

Integrate $f(x, y, z) = z^2$ over $A = \{(x, y, z) \in \mathbb{R^3} | x^2+y^2+z^2 \leq R^2, x^2 + y^2 + z^2 \leq 2RZ\}$ I know $A$ is the intersection between two spheres but I am unable to figure ...
0
votes
1answer
49 views

How to solve this integral in moment generating function

The moment generating function of generalised Pareto distribution eventually comes down to the following integral (here). $$ M_X(\theta) = \mathbb Ee^{X\theta} = \int_\mu^\infty e^{\theta ...
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0answers
11 views

Fundamental Theorem of Calculus for Line Integrals

Use the Fundamental Theorem of Calculus for Line Integrals to compute $\int_C F*dr$ where $$F(x,y,z)=(yz+2x)i+(xz+2y)j+(xy-2z)k$$ and C is the path from $(1,6,-1)$ to $(5,2,3)$ given by $x(t)=2t+1, ...
-2
votes
2answers
55 views

$\int _{k\pi }^{\left(k+1\right)\pi }\:\left|\sin\left(x\right)\right|dx$ [on hold]

How can I solve the following integral? $\int _{k\pi }^{\left(k+1\right)\pi }\:\left|\sin\left(x\right)\right|dx$
4
votes
1answer
79 views

Can $\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{(z-1)} dz$ be solved?

How we can calculate the result of following Integral? $$\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{z-1} \mathrm{d}z$$
0
votes
1answer
20 views

Leibniz rule for an improper integral

It follows from leibniz rule that if $\frac{\partial f}{\partial \theta_0}(\theta,\theta_0)$ exists then $$\frac{d}{d\theta_0}\bigg(\int_0^{\theta_0}f(\theta,\theta_0)d\theta\bigg)=\int ...
1
vote
1answer
65 views

I can't understand how can prove

I don't understand how we can prove that inequality, without integration $$\frac{1}{x}\int_{x}^{2x}(2-\frac{1}{y+2})\,dy \geq 2 - \frac{1}{x+2}.$$ P.S: Here is what I try... if can someone help me to ...
1
vote
1answer
27 views

calculate integral of given function

let us consider following integral while if we calculate from -infinity to plus infinity then it says that generally it should be 1/infinity +1/infinity right? which should be equal to ...
-1
votes
1answer
48 views

Nobody can help me ? I can't believe that…

Okay, we have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}$, and we need to prove that: 1)$I_n\le log\left(2\right)$ $,\:\:\:\:\:$ why just log(2) ? can not be 1? ...
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votes
1answer
56 views

integration of $e^{-\frac{t}{\tau}}$ [on hold]

$$\int_0^\tau e^{-\frac{t}{\tau}}\mathrm{dt}$$ Please give a very detailed explanation. Anyway, the answer to this is $\tau \left(1-\frac{1}{e}\right).$
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0answers
15 views

Solution $\frac{-1}{\mathrm{Beta}[a,b]\:l^r}\int_0^1 \ln\left(\frac{1-z^{1/c}}{1-z^{1/c}+dz^{1/c}}\right)^r z^{(ac-1)/c}(1-z)^{b-1} \, dz$

I need to solve the following integral $$\int_0^\infty x^r\frac{c}{\mathrm{Beta}[a,b]} \left(1-\frac{d\exp(-lx)}{1-(1-d)\exp(-l\: x)}\right)^{ac-1} ...
-1
votes
2answers
53 views

How to integrate $\frac{1}{(1+a\cos x)}$ from $-\pi$ to $\pi$ [duplicate]

How to solve the following integration?$$\int_{-\pi}^\pi\frac{1}{1+a \cos x}$$
0
votes
3answers
45 views

Derivative of a definite integral?

I could not figure out what I am doing wrong. Suppose $$f(x)= \int_1^x \sin(t^2) \ dt$$ What is $f'(x)$? I found $f'(x) = 0$. But it says this is not correct answer. Can someone please explain step ...
1
vote
1answer
73 views

Which of the following is true for $\int_{1}^{0} x\ln x\, \text dx$?

Which of the following is true for $\int_{1}^{0} x\ln x\,\text dx$ it is equal to $−1/4$ it is divergent it is equal to an irrational number does not have a closed form it is impossible to ...
0
votes
2answers
50 views

Need help with Riemann sum

Find $$\int_{a}^{b} x^m dx$$ where $0<a<b$ and $m\neq -1$. The answer goes as follows : but I get lost in the calculations which I need help with pinning down precisely: We choose the points ...
0
votes
1answer
37 views
1
vote
1answer
49 views

Taylor polynomial for an integral

This is the first time encountering a Taylor expansion along with an integral, so I am wondering how I should proceed. Question: $Consider \space the \space function$ $$F(x) = ...
1
vote
1answer
21 views

Integral using complex numbers shortcut

I want to compute the following integral $$- \frac{1}{M(\lambda_1-\lambda_2)}\int\limits_{-\infty}^t(e^{\lambda_1(t-t')}-e^{\lambda_2(t-t')})(\beta\omega A\sin\omega t' +g)\;dt'$$ Here the integral ...
0
votes
2answers
61 views

Why does the integral of absolute value function not return actual area?

Suppose my function is f(x)=$x^2-1$. The absolute value of this function is $\sqrt{(x^2-1)^2}$. So why doesn't the area of this function between -2 and 2 equal $\int_{-2}^2\sqrt{(x^2-1)^2}$? The ...
1
vote
0answers
34 views

Confused about integration over zeroes.

Does for example $\int_{-\pi}^{\pi} \sin(x) \, dx$ cancel out to zero (following WolframAlpha/normal integration technique), or do we have to take the absolute value of all the areas between bounds ...
0
votes
1answer
35 views

Continuous Annuity Question

I need to calculate the present value of a level continuous annuity which pays $1000/mo. for 10 years. The force of interest is 5/(3+2t). I tried taking the integral of e^(integral of force of ...
0
votes
3answers
58 views

Given several integrals calculate ${\int\limits_5^6}$ f(x) $dx$

Let $\int\limits_4^7 f(x)\,dx = 2$, $\int\limits_6^7 f(x)\,dx = 17$, and $\int\limits_4^5 f(x)\,dx = 3$ Calculate $$\int\limits_5^6 f(x)\,dx$$ I guess I am to assume that $f(x)$ is the same in all ...
1
vote
2answers
36 views

convert riemann sum $\lim_{n\to\infty}\sum_{i=1}^{n} \frac{15 \cdot \frac{3 i}{n} - 24}{n}$ to integral notation

The limit $ \quad\quad \displaystyle \lim_{n\to\infty}\sum_{i=1}^{n} \frac{15 \cdot \frac{3 i}{n} - 24}{n} $ is the limit of a Riemann sum for a certain definite integral $ \quad\quad ...