Questions about the evaluation of specific definite integrals.

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25 views

Solving an integral (with substitution?)

For a physical problem I have to solve $\sqrt{\frac{m}{2E}}\int_0^{2\pi /a}\frac{1}{(1-\frac{U}{E} \tan^2(ax))^{1/2}}dx $ I already tried substituting $1-\frac{U}{E}\tan^2(ax)$ and ...
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2answers
70 views

Evaluate the definite-integrals $\int_0^{\pi} \frac{\sin nx}{\sin x}dx$

Evaluate the definite-integrals $\int_0^{\pi} \dfrac{\sin nx}{\sin x}dx$ my teacher say that, using the formula : $\sin nt=\dfrac{e^{ni}-e^{-ni}}{2i}$, but i can't :(.
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0answers
19 views

Calculate the flux coming out of a surface

Let F(x,y,z)=(2xy(z-2),x^2(z-2),x^2y) be a vector field and $\Sigma $ the surface defined as the portion of cone x^2+y^2=(z-2)^2 ...
2
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0answers
28 views

Determine the volume of $A:=\{(x,y,z)\in \mathbb R^3 : \sqrt{x^2+y^2}\leq f(z)\}$

Let $f\in L^2(\mathbb R)$ and $f\geq0$. Determine $A:=\{(x,y,z)\in \mathbb R^3 : \sqrt{x^2+y^2}\leq f(z)\}$. "Normal" substitution $(x=rcos(\phi),y=rsin(\phi))$ did not help a lot, since I dont have ...
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2answers
58 views

Area enclosed by cardioid using Green's theorem

Let $$\gamma(t) = \begin{pmatrix} (1+\cos t)\cos t \\ (1+ \cos t) \sin t \end{pmatrix}, \qquad t \in [0,2\pi].$$ Find the area enclosed by $\gamma$ using Green's theorem. So the area enclosed by ...
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0answers
27 views

How do I verify that $\int_0^1 (1-t) \, f''(t) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$ [on hold]

How do I verify that: $$\int_0^1 (1-t) \, f''(ht+x) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$$
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29 views

Riemann integrable function to the power of $p \in [1,\infty[$ is R-int. again.

It is stated that it is sufficient to prove Riemann-integrability of $|f|^p$ for $0 \leq f \leq 1$. $(f,\psi,\varphi:[a,b] \rightarrow \mathbb R)$. $\checkmark$ For any $\varepsilon >0$ there ...
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1answer
23 views

Volumes of revolution?

The point $P(a,b)$ lies on the curve $y=\mathrm{arsinh}\,x$. $R$ is the region bounded by the curve, the $x$- and $y$-axes and the line $x=a$. When $R$ is rotated $2\pi$ radians about the $x$-axes the ...
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0answers
19 views

Volumes of revolutions question

The point $P(a,b)$ lies on the curve $y=arsinh x$. $R$ is the region bounded by the curve, the $x$- and $y$-axes and the line $x=a$. When $R$ is rotated 2$π$ radians about the $x$-axes the solid ...
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2answers
76 views

Why does $\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$?

I was wondering why the following is true: $$\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$$ It is easy to obtain this result by doing a trig substitution but it's messy and not ...
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0answers
33 views

Examples of double surface integrals

I'm looking for comprehensible introductions, detailed examples and practical applications of double surface integrals. I'm particularly interested in parametric surfaces and numerical integration ...
2
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1answer
28 views

Evaluate $\int^\infty_0 t^{a+b-1}(t+1)^{-b-1} U(a+2,a-b+2,ct)dt$

Evaluate $$ \int^\infty_0 t^{a+b-1}\left(t+1\right)^{-b-1} U\left(a+2,a-b+2,ct\right)dt $$ under the condition $a>0$, $b>0$ and $c>0$, where $U(\cdot,\cdot,\cdot)$ denotes the ...
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0answers
17 views

Find the Volume of the Solid--Cylindrical Shells

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $$ y = 2x^2, x = 1, y = 0 $$ about the x-axis I can't seem to get this. It's in a ...
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0answers
23 views

Calculate the arclength of $y=\frac{\sqrt{x}(x^2+2)}{3}, \hspace{0.5cm} 2\leq x\leq 4$

Calculate the arclength of $$y=\frac{\sqrt{x}(x^2+2)}{3}, \hspace{0.5cm} 2\leq x\leq 4$$ My work: I have calculated $$1+(\frac{dy}{dx})^2=\frac{25x^4+20x^2+36x+4}{36x}$$ I need your help to ...
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2answers
66 views

How to evaluate the integral $\int x^2/\sqrt{4-x^2}\,dx$?

How to compute this integral? $$\int \frac{x^2}{\sqrt{4-x^2}}dx$$ If there were $x$ instead of $x^2$ in the numerator I know how to do a substitution $y=4-x^2$. But this doesn't help with the $x^2$.
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1answer
56 views

How do I integrate $\int_0^{2\pi} [x\sin x]\,dx $, where $[\cdot]$ is the greatest integer function? [on hold]

Integrate $$\int_0^{2\pi} [x\sin x]\,dx, $$where $[\cdot]$ is the greatest integer function.
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1answer
55 views

Evaluating $\int_0^\infty dn \, \frac{x^n}{(3n+1)(3n+2)}$

I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate, $$\int_0^\infty dn \, ...
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0answers
66 views

Can we interchange the Integral and Summation when a limit is $\infty$?

I was trying to Evaluate the Integral: $$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$ $$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot ...
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5answers
104 views

Evaluating $\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx$

How do I evaluate the definite integral $$\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx ?$$ I used trig substitution, and then a u substitution for $\sec\theta$. I tried doing it and got an answer of: ...
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1answer
13 views

What part of area is included in the definite integral?

I am supposed to find the area of the blue shaded region. The line $\mathrm {OA}$ is $y=x$. The circle is $x^2+y^2=16$. The best method is to find slope of the line and use the formula ...
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3answers
20 views

Indefinite integrals with rati0nal and polynomial functions and Substituion

I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would ...
7
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2answers
165 views

how to compute this limit

compute $I=\lim\limits_{n\to+\infty}\sqrt[n]{\int\limits_0^1x^{n+1}(1-x)\cdots(1-x^n)dx}$ attempt: I tried to evaluate the integral $$\begin{align} ...
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3answers
110 views

Solving $\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$

Evaluate $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$$ This is my attempt: $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x} = ...
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3answers
62 views

Evaluation of integral $\int_{-\infty}^{+\infty} xe^{-|x|}\,dx$ is not $0$

Given $$f(x)=\frac12e^{-|x|}, -\infty \le x \le +\infty$$ $$\int_{-\infty}^{+\infty} x f(x)\, dx= -\frac12\int_{-\infty}^{+\infty} x (-e^{-|x|})' dx=-\frac12\bigg(-xe^{-|x|} + ...
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+50

analytic solution of a definite integral

Integrate the following $$\int_0^\infty \alpha\,\beta\, c\, k\, x^s\, x^{c-1} (1+x^c)^{k-1} \left[(1+x^c)^k-1\right]^{-\beta-1} \left[1+\gamma ...
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3answers
48 views

Integral $\int_{0}^{\infty}\frac1{\sqrt[\alpha]{1+x^\beta}}dx$

Is there a general answer for the integral of the form:$$\int_{0}^{\infty}\dfrac1{\sqrt[\alpha]{1+x^\beta}}dx$$
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1answer
59 views

Integral $\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$

How to evaluate: $$\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$$ Basically am evaluating value of $\log(1+c\chi^2)$ where $\chi^2$ is $\chi$-squared distributed
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1answer
40 views

When does the integral converges?

For what $\alpha, \beta$ the integral $$\int_0^\frac{\pi}{2} \frac{(\frac{\pi}{2} - x)^\alpha}{(\cos x)^\beta} dx$$ converges? So first I used WolframAlpha to know that $\frac{\pi}{2} - x < ...
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0answers
12 views

Calculation of probabilities in Z table

I would like to calculate at least one probability from z table. I know that pdf for N(0,1) is 1/(2*pi)*exp^(-(x^2/2)). Also the cdf is However, I do not know how to calculate this integral. ...
2
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1answer
26 views

How to proceed with the following integration?

If $n$ is a positive integer, show that $$ \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin(t^2) dt = \frac{(-1)^n}{c}$$ for some $c \in [\sqrt{n\pi}, \sqrt{(n+1)\pi}]$ I have an idea that i can use Mean ...
1
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1answer
16 views

Prove that for every $T > \frac{\pi}{2} $, $\int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx < 0$

I tried doing integration by parts a few times, after doing it 3 times I get the following expression: $$ \int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx = \frac{sin(T)}{T} - \frac{1}{\frac{\pi}{2}} - ...
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3answers
48 views

How to convert into a definite integral

Could you show me how to convert the following into a definite integral: $$\lim\limits_{n \to \infty} \sum_{k=1}^{3n} \frac{1}{n}\cos\left(\frac{k\pi}{n}\right)\sin\left(\frac{2k\pi}{n}\right)$$ ...
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0answers
29 views

Simplify $\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$

EDIT: I would love to find an analytical solution for this definite integral: $$\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$$ with $\delta>0,\, R>0,\,b>0$. Does ...
1
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1answer
30 views

Compute integral given 2 other integrals

I want to know which solution is correct. The question states: If f is an integrable function on [1,3], and if $$\int_1^2f(x)dx=4 \space\space\space\space\space\space and \space\space\space\space ...
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1answer
36 views

How to compute $\sum\limits_{a=1}^{\infty}\int_0^b\lambda\left(\int_0^{\lambda}e^{-t}t^{a-1}dt\right)d\lambda$

Please suggest an efficient method to compute the following integral \begin{equation} I = \sum\limits_{a=1}^{\infty}\int_0^b\lambda\left(\int_0^{\lambda}e^{-t}t^{a-1}dt\right)d\lambda \end{equation} ...
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2answers
16 views

How to prove $L_{f}(P) \leq L_{f}(Q)$ when $Q$ and $P$ are partitions of $[a,b]$ and $Q \supseteq P$

I'm having trouble proving this idea. Suppose that $f$ is bounded on the interval $[a, b]$. $P$ and $Q$ are partitions of $[a, b]$, and $Q \supseteq P$. $$ L_{f}(P) \leq L_{f}(Q) $$ I know that this ...
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0answers
20 views

Evaluating the surface integral side of a divergence theorem problem

Edit: I realized where my mistake was...thanks for the help! In class we were discussing a problem (page 1185 in Smith and Minton's Calculus Third Edition): Let Q be the solid bounded by the ...
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2answers
73 views

Unusual integral

I got a clock as a gift recently. It has a very novel face in that the hour positions are given by a complex formula. For the most part, I have been able to verify the calculations presented as ...
1
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1answer
16 views

Volume of Revolution Verification

Question: A region in the $xy$-plane is bounded by the $x$-axis, the lines $x=1$, $x=2$ and the curve $y=2x^2 +1$. Find the volume obtained by rotating the region around the $x$-axis. I did ...
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0answers
26 views

For the following integrals find a and find b [closed]

In the following picture, what is a=? what is b=?
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1answer
39 views

Evaluation of the integral $I=\int_0^{\infty}[1-J_0(kr)]k^{1-\alpha}dk$

How to compute \begin{equation} I(r) = \int_0^{\infty}[1-J_0(kr)]k^{1-\alpha}dk,\quad 2<\alpha<4 \end{equation} where $J_0(x)$ is the zeroth order Bessel function of the first kind. A paper ...
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1answer
51 views

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: ...
1
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1answer
40 views

Integral $\int_0^\infty \frac{e^{-cy} dy }{1+ay}$

$$I'=\int_0^\infty \frac{e^{-cy} dy }{1+ay}$$ a, b, and c, are positive coefficients. This integral is part of a problem which I'm trying to solve it and after lot's of effort the problem transform ...
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0answers
50 views

evaluating an indefinite and improper integral

$$ I=\int_0^{+\infty} \frac{Q_1(a,\sqrt{by})}{1+cy}dy $$ $a$, $b$, and $c$ are positive coefficients. $Q_1$ is Marcum $q$-function. This integral is part of a problem which I'm trying to solve it and ...
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1answer
81 views

Estimating the integral $\int \frac{\sin(x)}{x}\, dx$. [closed]

Would anyone be able to help me out with this question? I'm not quite sure how to go about it. Thanks in advance! Consider the integral $$ I = \int_{\pi/2}^\pi \frac{\sin x}{x}\,dx. $$ This integral ...
11
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4answers
165 views

How to compute $\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$

Could you explain to me, with details, how to compute this integral, find its principal value? $$\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$$ $f(z) =\frac{\sqrt{z}}{z^2-1} = \frac{z}{z^{1/2} ...
0
votes
0answers
33 views

Riemann sum/integral of $\sin(x)$ from $0$ to $A$ [duplicate]

Hello I keep getting stuck on calculating the Riemann sum/integral of $\sin x$ from $0$ to $A$ I know this has been looked at before but I just don't understand it and was hoping someone could ...
0
votes
1answer
35 views

Riemann Integral Property for Continuous, Monotonic, Non-negative Function

If $f$ is continuous, non-negative, and monotonically increasing function on $[0,∞)$, then prove that $\int^{x}_{0} f(t)dt\leq xf(x)$ $\forall x ≥ 0$ My attempt: Define $F(x)=\int^{x}_{0} f(t)dt$. ...
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votes
0answers
20 views

Definite integrals and piecewise defined functions [closed]

Consider the function $G(x) = \int_0^x g(u)\, du$ , where: $ g(u) = \begin{cases} 4 - \frac 43 u, & \text{for $0 \leq u < 6$} \\ u - 10, & \text{for $6 \leq u \leq 12$}. \end{cases} $ i. ...
5
votes
1answer
190 views

Why do we put absolute brackets for ln?

When writing out the final answer in $\ln$ form, why is it necessary to put absolute brackets? How does it affect the answer? I have this answer of $-3\ln|\frac{3+\sqrt{9-x^2}}{x}|$, but why does it ...