Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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1answer
10 views

Integration Convergence/Divergence Questtion

$$ \int\limits_0^{\pi} \frac{ dt}{\sqrt{t} + \sin t }$$ How can one tell if this integral converges or diverges? Integral of 1/(t^.5+sin(t)) from 0 to pi. I can't even find the antiderivative.
-1
votes
1answer
24 views

Prove that $\int_0^{\pi} \sin^nx\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cos(n+2)xdx=0$

Prove that $$\int_0^{\pi} \sin^nx\cdot\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cdot\cos(n+2)xdx=0$$ with $n \in \mathbb{N}$ I think it's true, but I can't prove.
2
votes
1answer
15 views

Integration by parts $\int(x+y)e^{-x}dx$

What I'm trying to solve: $\int(x+y)e^{-x}dx$ Here's my professor's approach: $$u = x, du = e^{-x}$$ $$du = dx, dv = -e^{-x}$$ By doing parts: $(-xe^{-x}) - \int(-e^{-x})dx - ye^{-x} = (-xe^{-x}) ...
3
votes
1answer
33 views

How to use U substitution for the integral $\int\frac{8}{49+x^2}\,dx$?

So, the following is my problem. $$\int\frac{8}{49+x^2}\,dx$$ I understand this. I should first take out the constant which is 8 so it'll be $$8\int\frac{1}{49+x^2}\,dx$$ Then I should factor out the ...
0
votes
2answers
17 views

Integration inequality question help: Sketch the curve y=1/u for u > 0…

Sketch the curve $y=\frac{1}{u}$ for $u > 0$. From the diagram, show that $\int\limits_1^{\sqrt{x}}\frac{du}{u}< \sqrt x-1$, for x > 1. Use this result to show that $0 < \ln(x) < ...
3
votes
1answer
61 views

Integration help - question: e^(-sin(x))

I would really like some help with the integration of e^(-sin(x)). Thanks you to anyone who will help :) Given that sin(x) > 2x/pi for 0 < x < pi/2, where $$\int_0^{\pi/2}e^{-\sin x}\,dx$$ ...
3
votes
1answer
24 views

Why does $\int_b^{b+\Delta b}f(x)\;dx=f(b)\Delta b+\mathcal{O}(\Delta b^2)$

On this page it is shown that: $$\frac{\partial}{\partial b}\left(\int_a^bf(x)\;dx\right)=\lim_{\Delta b\rightarrow 0}\frac{1}{\Delta b}\int_b^{b+\Delta b}f(x)\;dx=\lim_{\Delta b\rightarrow ...
3
votes
1answer
44 views

Why does the integral equal $1$?

Let $a\in\mathbb{R}-\mathbb{Z}$. Why is the following equality true? $$1 = \frac{1}{2\pi} \int_0^{2\pi} \left| e^{-i(\pi-x)a} \right|^2 dx$$ More precisely, why is the integrand equals $1$?
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1answer
21 views

help with improper integral and cuberoot? [on hold]

Evaluate the following integral . $$\int_{-\infty}^{0} \frac{1}{(x+2)^{1/3}}dx$$
0
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0answers
19 views

FP3 Integration help [duplicate]

$$I_{n}=\int x^n(1-x^2)^{\frac{1}{2}} dx$$ Show that $$(n+2)I_{n}=(n-1)I_{n-2}-x^{n-1}(1-x^2)^{\frac{3}{2}}$$ So far I have done this: $$\int x^{n-1}(x)(1-x^2)^{\frac{1}{2}} dx$$ $$u=x^{n-1}$$ ...
0
votes
1answer
34 views

Reduction formulae

$$I_{n}=\int x^n(1-x^2)^{\frac{1}{2}} dx$$ Show that $$(n+2)I_{n}=(n-1)I_{n-2}-x^{n-1}(1-x^2)^{\frac{3}{2}}$$ I can't seem to get this answer. Can someone please explain how to get to this? Thanks ...
1
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0answers
27 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 ...
3
votes
0answers
15 views

Understanding averaging of symplectic matrices via Haar measure

In McDuff and Salamon's Intro. to Symplectic Topology (2nd edition), there's a proof that $U(n)$ is a maximal compact subgroup of $Sp(2n)$ which I'm trying to understand. The proof uses the Haar ...
0
votes
1answer
18 views

Integral evaluation involving trignometric functions

How to explain the following equality? (Part of an integral calculation): $$\frac{2}{2\pi}\int_{-\pi}^\pi \left| \sin x \right| (\cos nx + i\sin nx) dx = \frac{4}{2\pi}\int_0^{2\pi} \sin x \cos nx ...
0
votes
0answers
15 views

Proof of the Poincare inequality for $W_0^{1,2}((a,b))$.

I have a question about an exercise for which I already have the solution, which I do not unterstand completely. Let $a, b \in \mathbb R$ with $0 < a < b$. Then we have \begin{align*} ...
3
votes
3answers
77 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
1
vote
1answer
27 views

Polygon Matching [on hold]

Given a set of polygons vertices and a template polygon vertices, find all that match polygons from the given set of polygons with a template polygon.
2
votes
2answers
25 views

Evaluate 2D integral (by change of variable)

The question asks to evaluate integral $$\iint_D \Big[3-\frac12( \frac{x^2}{a^2}+\frac{y^2}{b^2})\Big] \, dx \, dy \ $$ where D is the region $$\frac{x^2}{a^2}+\frac{y^2}{b^2} \le 4 $$ I believe ...
-5
votes
0answers
15 views

Checking Fourier Series Problem [on hold]

I just worked out this sum. Please check this derivation.
1
vote
2answers
40 views

Prove the relation $\frac{1}{x}$=$\int^\infty_0$ $e^{-xt}$ dt, for $x>0$. Use it to prove $\int^\infty_0$ $\frac{\sin(x)}{x}$ dx = $\frac{\pi}{2}$

Prove the relation $$\frac{1}{x} = \int^\infty_0 e^{-xt}\, \text{d}t, \text{ for } x>0.$$ Use it to prove $$\int^\infty_0\frac{\sin(x)}{x}\, \text{d}x = \frac{\pi}{2}.$$ "Hint: Use ...
0
votes
4answers
27 views

Integral with polynomial

I have a problem with this this integral. $\int \frac x{x^2+2x+2} \,dx$ I know that result is connected with logarithm, because the numerator is derivative of denominator, but i can't figure how to ...
0
votes
1answer
37 views

Mean Value Theorem for Integrals

I understand their is an easy way of doing this, I just want to check if my working to a more complicated method is correct. $f$ continuous on $[a,b] \implies $ Riemann Integrable on $[a,b]$ ...
0
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0answers
23 views

Understanding Green's Theorem

When looking at Goursat's theorem in complex analysis, I came across the Wiki proof which involves beautiful application of Green's theorem. I saw Greens theorem simply as "connection between line ...
2
votes
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 ...
2
votes
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 ...
0
votes
1answer
33 views

Trigonometric integral (arctg)

I have a problem with this integral. $$\int \text{arctan}(x-2)dx =\text{ }?$$ I tried integration by parts but it doesn't lead to right result.
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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: $$\int_0^1 (1-t) \, f''(ht+x) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$$
1
vote
4answers
24 views

How to caluclate the integral of $\int \frac{1}{\sqrt{4x^{2}+1}}dx$ using a trig substitution?

I am trying to determine the following integral: $\int \frac{1}{\sqrt{4x^{2}+1}} dx$ using a suitable substitution. My progress: let $x = \frac{1}{2} \tan \theta$ $dx = \frac{1}{2}\sec^{2} \theta ...
3
votes
0answers
22 views

Issues proving a basis via wedge product

On a quiz I was given the problem" a series that is a basis for $[-1,1]$ is $ \sum_0^{\infty} c_n P_n $, where $ P_n $ is a polynomial and each polynomial $P_n$ is orthonormal to the others. Using the ...
2
votes
2answers
68 views

How to solve such an integration analytically?

$\displaystyle\int^{2\pi}_{0} e^{ia \cos{\theta}}d\theta$ where $a$ is some constant. Can it be solved with some substitution? I tried it by expanding the exponential series but that was not proper ...
0
votes
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 ...
0
votes
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 ...
4
votes
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 ...
1
vote
2answers
57 views

Find $\int\frac{dx}{2+\sqrt{x}}$ (using Integration by Substitution)

I used the substitution: $u=x$ $du=dx$ $2+\sqrt{u}=2+\sqrt{x}$ I then substituted the u into the equation: $\int\frac{1}{2+\sqrt{u}}du$ $=\int{(2+\sqrt{u})^{-1}du}$ I'm not too sure how to ...
1
vote
2answers
34 views

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$ I've seen a proof which goes like this. $$ \lim_{x\to\infty} \frac{\frac{\arctan x}{x^2}}{\frac{1}{x^2}} = \frac{\pi}{2} > ...
2
votes
1answer
62 views

finding $\int\frac{1}{(t^2+25)^2} dt$ without trig substitution

Our calculus book covers partial fractions but not trig substitution, so I would like to find out the most elementary way to evaluate $$\displaystyle\int\frac{1}{(t^2+25)^2}\;dt$$ without using ...
1
vote
2answers
48 views

Integration of the following

What is the definite integral of $$ \int_0^1 \left(\frac{g(x)}{f(x)}\right)'\cdot\frac{1}{g(x)}\,dx, $$ where the conditions are as follows: $f(0) = 2 $ $f(1) = 3 $ $f'(x) $ is continuous For all ...
0
votes
1answer
53 views

Evaluate the integral $\int_1^\infty \frac{2^x}{2^{(2^x)}}dx$

Evaluate the integral $$\int_1^\infty \frac{2^x}{2^{(2^x)}}dx$$ My Try: substituting $t = 2^x$ we get: $$\ln 2 \int_2^\infty \left(\frac{1}{2}\right)^t dt = \frac{\ln 2}{\ln 0.5} \left( ...
3
votes
0answers
50 views

$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
2
votes
2answers
254 views

Trig substitution fails for evaluating $ \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx$?

Evaluate the integral \begin{equation} \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx \end{equation} Basically I could substitute: $t = \sin x$ and get: $$\int \frac{t}{t^2 +t + 1} dt$$ But, ...
1
vote
1answer
24 views

Sum of uniform random variables $U(0,1)$ and $U(0, a)$

The problem I have is: $X \sim U(0,1), Y \sim U(0,a)$ are independent random variables. Find the pdf of $X + Y$. I've got stuck in an integral-problem, and will show you what I've tried. Skip to the ...
1
vote
1answer
38 views

Why doesn't this method for integration by parts work?

So here is what I did first. $$∫16\ln(x^{1/3})dx$$ move the constant $16$ out $$16∫\ln(x^{1/3})dx$$ use properties of logarithms to rewrite natural log of cube root of $x$ as $\ln x$ divided by $3$ ...
0
votes
0answers
7 views

Transformation of the infinitessimal integration variable under a coordinate transformation

I always get confused when I'm facing the 3D integral over space and have to do a coordinate transformation on the given function to solve the integral. Do some of you have tips/trick how to ...
1
vote
1answer
21 views

Find all differentiable equations using Cauchy-Riemann equations

Let $z=x+iy$ and $f(z)=u(x,y)+iv(x,y)$. I want to use the Cauchy Riemann equations to find all differentiable functions of the form $$Re( h(z))=2x^2+2x+1-2y^2$$ So I used the C-R equations with ...
1
vote
1answer
53 views

Fundamental theorem of calculus, differentiable at the endpoints.

One version states: Let f be a continuous real-valued function defined on a closed interval $[a,b]$. Let f be the function defined for all x in $[a,b]$, by $F(x)=\int_{a}^xf(t)dt$. Then, F is ...
2
votes
0answers
11 views

Area of a region on the surface of a prolate spheroid

Is there a general expression for the area of a region bounded by 3 great ellipses on the surface of a prolate spheroid (where a great ellipse is the intersection of the spheroid with a plane passing ...
1
vote
2answers
30 views

How would I start to solve this?

I need to calculate the derivative of $F(x)=\int_{f(x)}^{f^2(x)}f^3(t)dt$. Usually for a derivative of an integral I would plug the upper bound and lower bounds into $f(t)$ then multiply each by their ...
2
votes
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 ...
1
vote
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 ...
0
votes
1answer
20 views

ODE: Why do we change our variable here?

I was trying to solve a matrix equation $\dot x = Ax + Bu$ Rearranging yields $\dot x - Ax = Bu$ Let $I = e^{-At}$ our integrating factor so $d(xe^{-At})/dt = e^{-At}Bu$ Then $xe^{-At}$ = $x_0 ...