Questions about the evaluation of specific definite integrals.

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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 ...
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40 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 ...
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3answers
78 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?
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4answers
95 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 ...
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1answer
48 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 ...
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2answers
83 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 ...
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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 ...
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2answers
28 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 ...
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2answers
20 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 ...
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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.
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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?
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2answers
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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 ...
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3answers
58 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 ...
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0answers
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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, ...
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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 ...
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$\int_{1}^{\infty}\frac {P(x)}{2x+1}\,dx = \frac {-3}{4}+\frac {1}{4}\log 2 + \frac {1}{2}\log 3$ [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 ...
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1answer
46 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 ...
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2answers
83 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} ...
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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 ...
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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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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 $x(t)=2t+1, ...
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2answers
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$\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
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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? $$\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{z-1} \mathrm{d}z$$
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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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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1answer
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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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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} ...
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How to integrate $\frac{1}{(1+a\cos x)}$ from $-\pi$ to $\pi$ [duplicate]

How to solve the following integration?$$\int_{-\pi}^\pi\frac{1}{1+a \cos x}$$
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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 ...
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1answer
72 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 ...
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2answers
49 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 ...
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1answer
37 views
1
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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) = ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
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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 ...
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1answer
63 views

how to compute the integral $\int_0^1 (1-x^p)^n dx$?

For constants $n$ and $p$, how to compute the integral $\int_0^1 (1-x^p)^n dx$ ? I saw a solution using hypergeometric function and another using incomplete beta function here: ...
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37 views

Integrate $sin(ax)/(x(x^2 + b^2)^2)$ from 0 to infinity, where a and b > 0. [closed]

Integrate the following function, where $a,b > 0$. $$\int_0^\infty\frac{\sin(ax)}{x(x^2+b^2)}\mathrm{dx}$$
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2answers
20 views

Calculate work done using integral

A chain lying on the ground is 10m long and its mass is 80kg. How much work is required to raise one end of the chain to a height of 6m? So im working in MKS system since its m and kg so the work ...
4
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2answers
273 views

How is this definite integral solved: $\int_{-\sqrt3}^{\sqrt3}{e^x\over(e^x+1)(x^2+1)}dx $?

$$\int_{-\sqrt3}^{\sqrt3}{e^x\over(e^x+1)(x^2+1)}dx $$ Tried partially integrating, had no luck.. Any thoughts?
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4answers
62 views

Finding the integral $\int_0^1 \frac{x^a - 1}{\log x} dx$

How to do the following integral: $$\int_{0}^1 \dfrac{x^a-1}{\log(x)}dx$$ where $a \geq 0$? I was asked this question by a friend, and couldn't think of any substitution that works. Plugging in ...
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2answers
34 views

Calculate $\int_\Gamma ze^{z}dz$ where $\Gamma$ is line from point $z_1=0$ to point $z_2=\frac{\pi i}{2}$

$$ \int_\Gamma ze^{z}dz\ $$ where $\Gamma$ is line from point $z_1=0$ to point $z_2=\frac{\pi i}{2}$ Hello, pls. how correctly calculate this example? I don't know what do with exponent..
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Find the closed form of $\int_0^{\large \frac{\pi}{2}}\frac{x^{2n}\cdot\log{{\sin{x}}}}{\sin^{2n}{x}}dx, \space n\ge 1$

I was thinking of the generalization of the problem here, that is $$\int_0^{\large \frac{\pi}{2}}\frac{x^{2n}\cdot\log{{\sin{x}}}}{\sin^{2n}{x}}dx, \space n\ge 1$$ Maybe you recommend me some tools? ...
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2answers
34 views

Compute $\lim_{x \rightarrow +\infty} \frac{[\int^x_0 e^{y^{2}} dy]^2}{\int^x_0 e^{2y^{2}}dy}$

I've tried to apply L'hopitals rule on this one, as this get's $\frac{\infty}{\infty}$ $$\lim_{x \rightarrow +\infty} \frac{[\int^x_0 e^{y^2}\mathrm{d}y]^2}{\int^x_0 e^{2y^2}\mathrm{d}y}$$ ...
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1answer
47 views

How do I integrate this

How do I perform this integration? $$\int_{0}^b x\frac{2b(b^2-x^2)}{(b^2+x^2)^2} dx = \ ?$$ I tried using integration by parts and arrived at an expression $$\int_{0}^{b} \frac {6b^3x^3 - ...