All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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Evaluate the following integral ∫4 sin^4 x cos^3 x dx

I can do simple integration problems, but problems like this seem to stump me, I created this problem so I could solve and compare it to another similar problem for my study guide, where should I ...
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7 views

Helffer-Sjöstrand-Formula: Idea behind?

I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor $\chi(y\langle x\rangle^{-1})$ for some bump function $\chi$ and the chinese symbol ...
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1answer
25 views

Initial Value Problem: $\frac{d^2y}{dt^2}=3-e^{2t};\;y(1)=-1;\;y'(1)=0$

$$\frac{d^2y}{dt^2}=3-e^{2t};\;y(1)=-1;\;y'(1)=0$$ I have never done initial value problems before, and this is very similar to one problem that is on my homework, I am trying to solve this to get a ...
2
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1answer
74 views

$\int_0^1\frac{1-t}{(t-2)\ln t}\,dt$ integral

I have two related questions. The first is: Is there a closed form expression for: $$\int_0^1\frac{1-t}{(t-2)\ln t}\,dt\approx0.507834$$ I know that there are some very superb integrators on this ...
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60 views

Help computing integral of quarter contour

For a project I'm working on I need to solve the following integral. Can anyone help me tackle this challenge? The Integral: $$ \beta \int_{\pi}^{\pi/2} {i k^2 r e^{i \theta} + ikr^2 e^{2i\theta} ...
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2answers
69 views

Using the substitution rule to integrate $ \int_{-1}^{2} (1+4x^2)\,dx $

I am stuck on an integral problem that came out of an arc length problem. I have an integral, $$ \int_{-1}^{2}1+4x^2dx. $$ When I try to apply the substitution rule, I am left with a variable that I ...
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1answer
20 views

How to find the volume of the region bounded by three graphs?

The problem asks given the region bounded by the graphs of y=lnx, y=0, and x=e, find a. the volume of the solid generated by revolving the region about the x-axis. b. the volume of the solid ...
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17 views

Help with calculating line integrals and potential functions [duplicate]

May you please help me with this questions? 1) Among all smooth, simple closed curves in the plan, oriented counterclockwise, find one along which the work done by the following vector is greatest: ...
2
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1answer
79 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
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1answer
60 views

Evaluate $\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx $

Evaluate the indefinite integral $$\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx$$ I tried multiplying and dividing by $\sec^2 {x} $ and then setting $\tan{x}=y$ but no good. Then I set $\cos ...
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3answers
34 views

line integral - what kind?

I need to calculate $\int_\Gamma F(x) \, dx$ from $(0,1)$ to $(-1,0)$ of the unit circle. $$F(x) = (x,y)$$ Now the answer is: But I don't understand what they did. Why $\Gamma(t) = (\cos t, \sin ...
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1answer
68 views

Integration by substitution doesn't work … Why?

I had this problem in my Math workbook: $$\int_0^{52.95}\sqrt{100-84\exp(-0.016t)} \, dt $$ I took a substitution of $u = 100-84\exp(-0.016t)$ such that $dt = 0.744\exp(0.016t) du$ After ...
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32 views

potential functions and line integrals

May you please help me with this questions? 1) Among all smooth, simple closed curves in the plan, oriented counterclockwise, find one along which the work done by the following vector is greatest: ...
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1answer
65 views

Analytical evaluation of the integral

Is there any analytical way to prove that the integral $\int_{2.04}^\infty \frac{\sin x}{x^2}dx$ is nonegative? I tryed to use geometrical approach, i.e. the graph of the integrand look like: The ...
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1answer
43 views

line integral…

Calculate $$\int_Γ f \, d\ell$$ for $f(x,y) = y, \; y=x^{1/2}$, $ x $ is in $[2,6]$. I know (now) that it means that: $$\int_\Gamma f \, d\ell=\int_a^b f(\Gamma(t)) \cdot \|\dot\Gamma(t)\| \, dt$$ ...
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2answers
22 views

Probability that the call will be answered at time $t$ is given by $f(t)$. Find the median waiting time for the call.

$$f(t) = \begin{cases} 0 & \text{if $t < 0$ } \\ 0.2e^{-t/5} & \text{if $t\geq 0$} \end{cases}$$. $ $ Find the median waiting time for the call. $ $ I cannot understand ...
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2answers
27 views

Integral related to the gamma function: $\int_0^{\infty} y^{\alpha-1}e^{-y}dy=(\alpha-1)\int_0^{\infty} y^{\alpha-2}e^{-y}dy$

I have difficulty with the gamma function: $\int_0^{\infty} y^{\alpha-1}e^{-y}dy=(\alpha-1)\int_0^{\infty} y^{\alpha-2}e^{-y}dy$ How do we go from left to right? Thanks!
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2answers
21 views

Integral of a sum of complex exponentials

Let $$\hat{\varphi_n}(t)=\frac{1}{n}\sum_{j=1}^n{exp(i{t}Y_j)}\quad(t\in\mathbb{R})$$ denote the empirical characteristic function of the residuals $Y_j\,=\,S_n^{-\frac{1}{2}}(X_j-\bar{X}_n),\quad ...
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1answer
33 views

An inequality concerning the relation between Riemann integral and the discrete sum

Prove that for all $n\in\mathbb{N,}\ n\geq 1,$ $$\frac{1}{2}\leq\sum_{k=1}^n\frac{1}{k}-\ln(n)\leq\frac{3}{2}.$$ My attempt: Regard $\sum_{k=1}^n\frac{1}{k}$ as the lower integral of the function ...
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2answers
37 views

Definite integral of function inverse to a polynomial

$f(x) = x+ x^3$ $g(y) = f^{-1}(x)$ $\int_2^{10} g(y) dy = $ ? $\int_2^{10} g(y) dy =$ [$dy = df(x), dy = 1+3x^2$] = $\int_1^2 (1+3x^2)dx$ = [$x+x^3$] =$ 10 - 2 =8$ Am I right or I did a mistake? ...
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33 views

ODE with multiple simple conditions $f'(x)=f(x)(Ax+D ) $

I have an ODE to solve . The main issue is,in addition to solving it I have to keep some conditions too in the solution of f(x).. I am bit confused regarding how to deal with it. Equation is given ...
3
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38 views

A probabilistic integral $\int_{-\infty}^{\infty}e^{-x^2/2\sigma^2}\arcsin\left(1-2\left|\lfloor x\rceil-x\right|\right)\,dx$

In my probabilistic studies, a tough integral appeared. Note that $\lfloor x\rceil$ is not the floor function; it is the nearest integer function. Up to some constants, it appears in a Buffon-like ...
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2answers
88 views

Integration of $x\cos(x)/(5+2\cos^2 x)$ on the interval from $0$ to $2\pi$

Compute the integral $$\int_{0}^{2\pi}\frac{x\cos(x)}{5+2\cos^2(x)}dx$$ My Try: I substitute $$\cos(x)=u$$ but it did not help. Please help me to solve this.Thanks
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1answer
94 views

How do you write an integral and why

A. Year 1 Calculus Student Approach $$ F(x) = \int f(x') dx\, $$ B. Random math paper you find online approach $$ F(x) = \int dx f(x') \, $$ C. Spivak $$ F(x) = \int f(x) \, $$ D. ??? (Edit) ...
9
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1answer
133 views

An integral with $e^{1+e^x}$ I had trouble working through

I had an analysis test earlier this morning and came across this integral, which I couldn't figure out. Parts of it are easy, but after integrating $y$ you're left integrating $xe^{1+e^x}$ which had ...
3
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1answer
44 views

Volume of revolving $ y = \sin(x) $ about a line $ y = c $

Consider the surface formed by revolving $y=sin(x)$ about the line $y=c$ from some $0\le{c}\le{1}$ along the interval $0\le{x}\le{\pi}$. Set up and evaluate an integral to calculate the volume V(c) ...
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4answers
130 views

Easier way to solve $\int\frac{2+\sin x}{\sin x(1+\cos x)}dx$

Is there easier way than universal supstitution to solve this integral $$\int\frac{2+\sin x}{\sin x(1+\cos x)}dx$$?
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1answer
34 views

Volume of the solid obtained by rotating half-disk around an axis

Consider the portion of the Cartesian plan delimited, in the first quadrant, from $x=0$, from $y=0$ and from the circumference of radius $= 1$ with center in the point $(0; 1)$, and determine the ...
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67 views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
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1answer
35 views

Volume of the solid with given base, whose sections with the planes orthogonal to $y = 0$ are rectangles of height $4$

Please help me to solve the following problem: Determine the volume of the solid having as base the portion of cartesian plane limited by $y = 0$ and by $y = x^{3}$ in the closed interval ...
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1answer
33 views

Question about the Fundamental Theorem of Calculus

So I have studied the FOTC, but not really sure of what I read so this question is just to help me learn the FOTC and understand how to do problems like it. $$ if $$ $$F(x)=\int_0^x\sqrt{sin^3(t)}dt$$ ...
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48 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
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49 views

How can you explain implicit differentiation?

So I am taking calculus 1 online from a local college (bad idea, but the only thing that fit my schedule). The professor used the notation $f'(x) =$ for EVERY function up until two weeks ago. All of ...
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1answer
287 views

Can all integration be thought of as projections?

For example, the integral of the function f(x) could be thought of the projection of f on the function g, where g is identically 1. Following this logic, can we think of the multiplication of f and ...
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24 views

How to evaluate the following integral? $∫_{-β}^{2π-β}\exp⁡(ix\cos(φ-β))dφ.$

I'm trying to calculate the following integral: $$∫_{-β}^{2π-β}\exp⁡(ix\cos(φ-β))dφ.$$ I tried by parts with no success and also by writing $\exp (ix)$ in terms of $\sin$ and $\cos$, with no ...
3
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1answer
66 views

Evaluating $\int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$

I'm trying to find the general formula for the following: $$I_n = \int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$$ I remember doing it a while back but for the life of me, I can't remember right now. ...
3
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0answers
78 views

Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
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11 views

Heuristic Algorithm for integrating algebraic functions

Is there any heuristic "algorithm" or a good technique for integrating algebraic functions? The general algebraic case was solved by Trager and Davenport. But their algorithms are rather complicated ...
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2answers
42 views

Help on an integration by substitution

In a proof to show that $\int_{0}^{1} f \left(\left\{1/x\right\}\right) \frac{ \mathrm{d}x}{1-x}=\int_{0}^{1} f(v) \frac{ \mathrm{d}v}{v}$, i found this line : ...
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22 views

what are and why are sine and cosine modulated integrals used?

I have found the definition of the following formulas in a paper, where they are called sine and cosine modulated integrals. $y$ is a signal with a strong periodic component of frequency $N\Omega$ ...
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13 views

Simplification of an integral comprising of vector-variables

How can I evaluate the simplify the integral $\int \rho (\bf{r^{\prime}})\, \delta (\frac{\sigma}{2}-r) d \bf{r^{\prime}}$ where $\delta$ is the dirac-delta function given that $\rho$ is constant ...
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3answers
92 views

Integral of $\int_{0}^{y} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x$

Does the following integral have a closed form solution? $$ \int_{0}^{y} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x $$ Or is there an approximation which works for large $\alpha$?
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1answer
56 views

Integration of powers: nested $dx$?

How do we solve the likes of the following expression: $$ \int_0^2 \frac{x\,dx}{\sqrt{1 + 2x^2}}\,dx $$ I'm bothered by the nested $dx$ in the numerator. How is this solved using the general power ...
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1answer
48 views

Is it true that $\int_{-C} f(x, y)ds = -\int_C f(x, y) ds$ [on hold]

I think it is more of a convention question, right ? $$\int_{-C} f(x,y)ds = -\int_C f(x,y) ds$$
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109 views

Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also ...
3
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2answers
164 views

Integral of exponential with $x(1-x)$ term

Does the following integral have a closed form solution? $$ \int_{0}^{y} \exp\left(\,\sqrt{\,x(1-x)\,}\,\right)\,{\rm d}x $$ Or must I settle with an approximation? Edit: Actual form of integral ...
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1answer
90 views

How to find the derivative of the function $ \int_{x}^{x^2}\frac{t}{\ln(t)}dt$? [on hold]

The problem is to find $\displaystyle\frac{d}{dx}\int_{x}^{x^2}\frac{t}{\ln(t)}\,dt$ I could do this if I had the first clue on how to integrate $\dfrac{t}{\ln(t)}$ but even wolframalpha is giving ...
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2answers
28 views

Finding the Value of K in an Integral Function

Given the function $$f(x)\begin{cases} -2(x+1), & \text{x $\le0$} \\ k(1-x^2), & \text{x $\gt0$} \\ \end{cases}$$ Find the value of k for $$\int_{-1}^1f(x)dx=1$$ Wasn't really sure how to ...
3
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4answers
127 views

How would I go about evaluating $\int \frac{x}{(9-8x^2)^3}dx$?

So I have homework on webAssign (a site used by my college), and I am not understanding the logic as to why I am taking the steps into solving the integral it is telling me to take. So I'll list the ...
1
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0answers
14 views

Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...