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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Analysis of Integral of a continuous function

one more question today I've been thinking on... Prove that if $f$ is continuous on $[a,b]$, $0<a<b$, then $\int_{a}^{b} {f(t) \over t} dt$ $=$ $\int_{a}^{s} {f(t) \over a} dt$ for some $s \in ...
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21 views

Integration Question

If we know the integral $$\int \frac{\mathrm{d}x}{f(x)+1}$$ can we find the integral of $$\int\frac{\mathrm{d}x}{f(x)+c}$$ for arbitrary $c\in\mathbb{R}$ (where defined)? Does it make a difference if ...
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2answers
30 views

Evaluating $\int \frac{1}{9+4x^2} dx$

Evaluate $$\int \dfrac{1}{9+4x^2} dx$$ I let $u = 9 + 4x^2$ so $du$ would be $8x$. But I don't know any way to make the numerator $1$ become $8x$. I could multiply by $1/8$ but then I'd still ...
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Meaning of Normal Vector in Surface Integration

Is there a good interpretation of what the normal vector (and its magnitude) $$\mathbf{N}=\frac{\partial \mathbf{X}}{\partial s}\times\frac{\partial\mathbf{X}}{\partial t}$$ to the parametric surface ...
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4answers
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Evaluate the integral of sec(2x + 1) dx

I got $\ln|\sec(2x +1) + \tan(2x+1)| + \text C$ as an answer. I saw that the integral of $\sec x$ is $\ln|\sec x + \tan x| + \text C$. But I feel I may have left something out because that was too ...
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1answer
19 views

Evaluate the integral $(3x-2)/(x+1)$

The answer I have which I'm sure is wrong is $(3x-2)\ln(\left \vert x+1 \right \vert + \text{constant}$ I let $x + 1$ be $u$ and $du$ would be $1$. But I didn't know how to get $1$ on top so I just ...
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2answers
33 views

Evaluate the integral $\int\frac{x^2 + 1}{x^3 + 3x + 1} dx$

Evaluate the integral $$\int\frac{x^2 + 1}{x^3 + 3x + 1} dx$$ I've looked at similar examples online and I can't find one like the one above. In class we did one where we had to do long division and ...
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1answer
17 views

Integral of cumulative normal

Let $$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} \exp\left({-\dfrac{\omega^2}{2}}\right) d\omega.$$ Question: for what values of $a$, $b$ and for what choices of $f(x)$ would the following ...
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1answer
11 views

Application of double integrals problem?

The centroid of a uniform plane region is at $(0,0)$ and the region has total mass $m$. Show that its moment of inertia about an axis perpendicular to the $xy$-plane at the point ($x_0$,$y_0$) is ...
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47 views

How to find $\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}$?

find the integral of $f(x)=\frac1{(x-1)^2\sqrt{x^2+6x}}$ my attempt = $(x-1)=a$, $a=x+1$ so the integral'd be $\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}=\int\frac{da}{a^2\sqrt{a^2+8a+7}} $ lets ...
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Integration involving complicated exponential form

I'm trying to simplify the following: $\int_0^t s^{-3/2} e^{-(a+bs)^2/(2s)} ds$ Basic substitution always gives a $s^{-1/2}ds$ counterpart which I don't know how to get rid of. Is there anyway to ...
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4answers
83 views

Antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$

What is the antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$ ? $\displaystyle\int t^2e^{-\frac{1}{2}t^2}\,dt=\displaystyle\int{t}_{}te^{-\frac{1}{2}t^2}\,dt=-te^{-\frac{1}{2}t^2}\Big|_{?}^?+\int ...
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1answer
15 views

Apply the Fourier Transform to $A\cdot e^{-a|k - k_0|}$

I have the following problem: The task is to show that $$f^*(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(k) e^{ik(x-vt)} dk$$ with $f(k) = A\cdot e^{-a|k - k_0|}$ equals $$f^*(k) = ...
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40 views

Evaluating the integral $\int\frac{3x^{2}-x+2}{x-1}\;dx$

As the title suggests, the following integral has been given to me $$\int\frac{3x^{2}-x+2}{x-1}\;dx$$ Yet I still get the wrong answer every time. Can someone calculate it step-by-step so I can ...
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Choosing the integral limits for marginal distribution

I've been trying to understand the following: The distribution of two continuous random variables is given by $$f_{X,Y}(x,y)=\frac{3}{7}x\space\space 1\le x\le 2,0\le y\le x$$and $0$ otherwise. ...
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2answers
31 views

Recursive formula for definite integral

The integral is: $$I_n = \int_0^{\pi/4} \tan^{2n}x\,dx$$ I'm supposed to find the recursive formula.
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4answers
55 views

Why is $ \int \frac{\sin x (b-a\cos x)}{(b^2+a^2-2ab \cos x)^{3/2}}\,dx = \frac{a-b\cos x}{b^2 \sqrt{a^2-2ab\cos x + b^2}}$?

Why is $$ \int \frac{\sin x (b-a\cos x)}{(b^2+a^2-2ab \cos x)^{3/2}}\,dx = \frac{a-b\cos x}{b^2 \sqrt{a^2-2ab\cos x + b^2}}\text{ ?}$$ Constant of integration omitted.
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1answer
14 views

Integreal around a unit circle

I know that when $m \in \mathbb{Z} \backslash \{ 0 \}$, we have $$ \int_0^1 e^{2 \pi i m \beta} \ d \beta = 0. $$ I was wondering if there is a simple formula for the following similar integral, when ...
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775 views

How Can I Attack This Integral?

$$ \int \frac{\sqrt{9-4x^{2}}}{x}dx $$ How Can I attack this kind of problem?
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1answer
36 views

$\int_{-1}^{1} x^{k+i} P_n(x)dx$, $P_n$ Legendre polynomial.

I was wondering whether there is a way to say what $$\int_{-1}^{1} x^{k} P_n(x)dx$$ is, where $k,n$ are positive integers or zero and $P_n$ is the n-th Legendre polynomial? I am looking for an ...
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1answer
26 views

Cases of Partial Fraction Decomposition

How many cases are there in integration using partial fractions?
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21 views

Integrable functions and absolute values

I have qutoted that the absolute value of an integral is less than or equal to the integral of an absolute value of a function. I have also said $|-g(x)| \le g(x) \le |g(x)|$ implies the integral ...
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1answer
21 views

product of bessel function integral

Is this formula is true for s=-1
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1answer
33 views

The volume of a cone whose length of its side is R

How can i compute the volume of a cone whose length of its side is $R$ and the vertex of the cone forms an angle $2 \theta$ . The top cone is a cap of a sphere of radius $R$. Some help please.
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40 views

How to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$?

Could you please give me some hint how to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$ when f(x) is continuous for $0<x\le1$ and $\lim_{x\to0^+}x^3f^2(x)=1$ ? I tried the usual way: ...
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62 views

Delicate Integral $I=\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set a=0 we get a similar integral given by $$ ...
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Fancy Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
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1answer
39 views

Integrating Differential Forms

This is part of a homework problem. I want to actually solve it myself, so no solutions, please (although this isn't even the full problem statement). I don't have a very good grasp on differential ...
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32 views

Area interpretation of integrals

When integrating under part of a circle, as in $$A=\int_0^a {\sqrt{r^2-x^2}\,\mathrm{d}x}$$ I noted that the simple geometric solution would be to add the areas of the sector and triangle formed by ...
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1answer
45 views

Sum as an integral

Recently I have encountered weird notation that I don't see into. When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this ...
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16 views

Definite integral of greatest integer function

I need to find the area under a function modeled by $f(x)=\left\lfloor 2.4x \right\rfloor+5$. I can't seem to figure out what the antiderivative of this is, so I'm going try to use a right Riemann ...
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1answer
22 views

Let $\int_a^bf(x)sgn(f(x)) + 2f(x) \ dx = 0$. Show that $f$ has at least one root.

The Assignment: Let $a,b \in\mathbb{R}$ and $a < b$. Furthermore let $f: [a,b] \rightarrow \mathbb{R}$ be differentiable and $|f(x)| + |f'(x)| \neq 0$ for $\forall x \in [a,b]$. Now, let ...
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27 views

An integral involving two variables and the floor function

Let $N$ be some fixed positive integer. I have the following function $$ g(z) = z \int_1^N [t] e^{2 \pi i t z} \ dt. $$ How would one compute $$ \int_0^1 g(z) \ dz ? $$ Thanks!
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1answer
30 views

Limits of integration

Is there any difference between the $$\int_a^b f(x) dx $$ and $$\lim_{x\to b^-} \int_a^x f(x) dx \qquad \text{OR}\qquad \lim_{x\to a^+} \int_x^b f(x) dx$$ When would one need the second versions of ...
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3answers
81 views

Evaluate $\displaystyle\int \frac{1}{1+x}\, dx$

I forgot about integrals so I need some help in this problem $\displaystyle\int \frac{1}{1+x}\, dx$ please.
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1answer
16 views

Showing that this function is not riemann integrable.

Consider the function h defined by h(x) := x+1 for x an element of [0,1] rational, and h(x) := 0 for x an element of [0,1] irrational. Show that h is not Riemann integrable. The hint in the back of ...
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3answers
71 views

Show $\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$

Assume $f$ and $g$ are monotonically increasing on $[0,a]$, Show that $$\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$$ If I differentiate both sides w.r. to $a$ then; ...
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2answers
19 views

Finding an integral $\int g(x)^j dx $ from $\int g(x)^2 dx $

let $I = \int_0^1 g(x)^2 dx $, where $g$ is a real valued function. With this information is it possible to give an upper bound for $\int_0^1 g(x)^j dx $? Here $j$ is a natural number. When $j=1$ I ...
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49 views

How to show that $f$ is Riemann integrable

Let $u,v:[a,b]\rightarrow\mathbb{R}$ be contunious. Define $f:[a,b]\rightarrow\mathbb{R}$ by $$f(x) = \begin{cases}u(x) & x \in \mathbb{Q} \\ v(x) & x \in \mathbb{R}-\mathbb{Q}\end{cases}$$ ...
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How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$?

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$ ? If I take $y=\sin\left(\frac{x}{2}\right)$ then, $\displaystyle ...
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1answer
33 views

Definition of the integral of a vector field on Riemannian manifold and Euclidean spaces

Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$? What if $M$ is a Euclidean space? Clearly the definition ...
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39 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
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41 views

What is the answer to $\int x(t)dt$?

$\int x(t)dt$? I'm trying to solve a differential equation, but I've hit a strange brick wall that I never used to have a problem climbing over. This question is about mechanics & the equation ...
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20 views

How to evalute $\displaystyle\int_{2}^{x}\frac{t\ dt}{(\ln^{m} t)(\ln^{n} (t+2))}$

How to evaluate this integral? $$\int_{2}^{x}\frac{t\ dt}{(\ln^{m} t)(\ln^{n} (t+2))}$$ Where $m$ and $n$ are positive integers.
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1answer
14 views

Volume between paraboloid and plane

I need to find the volume of the finite region enclosed between the surface $$ y = 1 - x^2 - 4z^2 $$ and the plane $$y = 0$$ Here's what I've done: $$ \int\int ...
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1answer
34 views

Integral inequality with first two moments equal to $1$.

Let $f\in \mathcal{C}^0([0,1],\mathbb{R})$ such that $$ \int_0^1 f(x)\text{d}x = \int_0^1 xf(x)\text{d}x=1.$$ Show that $\int_0^1 f(x)^2 \ge 4$. I tried to use Cauchy-Schwartz inequality such that ...
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97 views

Why is $\displaystyle\int^{\infty}_{0}{(1-\cos x)\over{x^{2}}}dx = \frac\pi{2}$?

I have been having trouble understanding Fourier series and analysis in one of my classes. This is one problem from the text and we have to show that this is true. I have done other problems related ...
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1answer
34 views

Support of $L^p$ functions?

I noticed something strange. If we look at a function $f \in L^p$, then this is an equivalence class. Hypothetically: $\operatorname{supp}(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is ...
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102 views

Some users are mind bogglingly skilled at integration. How did they get there?

Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this. Especially in a ...
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25 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...