Questions about the evaluation of specific definite integrals.

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Surface Area by Integration

$$2\pi\int_{3}^6\left(\frac{1}{3}x^\frac{3}{2}-x^\frac{1}{2}\right)\left(1+\left(\frac{1}{2}x^\frac{1}{2}-\frac{1}{2}x^\frac{-1}{2}\right)^2\right)^\frac{1}{2}dx$$ I've managed to simplify this down ...
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2answers
47 views

Evaluating the Definite Integral $\int_0^{\pi}\cos^{2n} \theta d\theta$

$$\int_0^{\pi}\cos^{2n} \theta d\theta$$ $$u=\cos \theta \implies du= -\sin \theta d\theta \implies d\theta= -\frac{du}{1-u^2} $$ $$\int_{-1}^1 \frac{u^n}{1-u^2} du=\int_{-1}^1 ...
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2answers
34 views

Finding the value of $3(\alpha-\beta)^2$ if $\int_0^2 f(x)dx=f(\alpha) +f(\beta)$ for all $f$

Let $f$ be a polynomial of degree $n$ at most $3$ such that there exists some $\alpha,\beta$ satisfying $\int_0^2 f(x)dx=f(\alpha) +f(\beta)$ for all such $f$. Find the value of $3(\alpha-\beta)^2$ ...
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1answer
37 views

Find $\int_0^1 \frac{dx}{(1+x^n)^2\sqrt[n]{1+x^n}}$

Find $$\int_0^1 \frac{dx}{(1+x^n)^2\sqrt[n]{1+x^n}}$$ with $n \in \mathbb{N}$ My tried: I think that, needing to find the value of $$I_1=\int_{0}^1 \frac{dx}{(1+x^n)\sqrt[n]{1+x^n}}$$ because: ...
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2answers
34 views

Integral $\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$

Is this integral known to have a closed form? $$\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$$ Is there anything special about it?
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2answers
83 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
74 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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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 ...
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29 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 ...
2
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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
24 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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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$Ļ€$ radians about the $x$-axes the solid ...
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2answers
77 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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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 ...
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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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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
59 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
56 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
67 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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105 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 ...
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2answers
169 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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1answer
101 views
+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
60 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
13 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
27 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 ...
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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 ...
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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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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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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 ...
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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. $$\lim_{Nā†’āˆž}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: ...
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1answer
41 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 ...
0
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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 ...