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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1
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3answers
38 views

convergence of $\int_a^b \frac{1}{x^2} dx$

Why is it true that $\int_0^a \frac{1}{x^2} = \infty$ but $\int_a^\infty \frac{1}{x^2} \lt \infty$? Shouldn't it be symmetric?
1
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1answer
21 views

Solving integral (by substitution?)

How do I solve the integral $\int \frac{1}{\sqrt{b-x^2}}$ where b is a constant ? I know that $\int \frac{1}{\sqrt{1-x^2}} = \arcsin(x)$ , so I guess I have to substitute somehow clever. Can you ...
0
votes
0answers
12 views

Simple proof for a continuous-time linear system and impulse $\delta$?

From Schaum's Outlines of Signals & Systems: Let's work with continuous-time signals. Let $T$ be a linear time-invariant system (LTI). Input $x(t)$ can be expressed as $x(t) = ...
1
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1answer
24 views

How one can compute $\int_0^1 \frac{dt}{t} t^{\pm \frac{1}{2}} \exp[ -x(t+t^{-1})] = \sqrt{\frac{\pi}{x}} e^{-2x}$?

For positive $x$ \begin{align} \int_0^1 \frac{dt}{t} t^{\pm \frac{1}{2}} \exp[ -x(t+t^{-1})] = \sqrt{\frac{\pi}{x}} e^{-2x} \end{align} How to compute this integral?
-1
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1answer
27 views

Divergence and convergence of the integral. [on hold]

I have the following integral, $$I=\int_a^b |x|^{-p} dx$$ where $a<b$ are finite real numbers and $p\leq 0$ is a non-negative real number. How one can determine $a, b$ and $p$ such that 1) I ...
2
votes
2answers
34 views

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$ Why is it wrong to... $$\int_0^{\ln(2)} \sqrt{(e^x-1)} dx= \int_0^{\ln(2)} (e^x-1)^{1/2} dx= \frac{2}{3}(e^x-1)^{3/2} |_0^{\ln(2)}$$
0
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0answers
23 views

Proof that there is no identity to integral operation on any set of functions

The statement is: Let $f\in F, f:x\mapsto f(x)$ be a function($F$ contains sufficiently non-trivial functions). Then $\not\exists I\in F$, so that $$\int_{-\infty}^\infty If=f(0)$$ What I am implying: ...
0
votes
0answers
32 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
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2answers
65 views

how to evaluate the integral $ \int_0^{2\pi} \frac{\sin{nx}\cos{nx}}{\sin{x}}dx$?

would someone give me a hint or a solution ? how to evaluate the integral $ \int_0^{2\pi} \frac{\sin{nx}\cos{nx}}{\sin{x}}dx$? Thanks a lot.
4
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0answers
24 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
0
votes
0answers
49 views

Integrate $x e^{-x^4}$ Advanced Calculus

I'm having trouble integrating the equation below. The book says "Hint: Let $u=x^2$, etc." I don't know how I would let $u=x^2$ since in order to do integration by parts I need $u\,dv$ and we only ...
0
votes
2answers
91 views

Find the antiderivative of $(x^2+x+1)^{20}$ [on hold]

How do I find the antiderivative for that? The online calculators say that there's no solution. **NVM, the little scratch on my paper turned out to be a faintly copied handwritten $'2'$, turning it ...
-2
votes
0answers
20 views

Definite integrals and piecewise defined functions [on hold]

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. ...
1
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2answers
104 views

For what values is my integral diverging or converging?

Is the following integral convergent $$\int_{\gamma}^{+\infty} \left(1-\dfrac{1}{1+sv^{-1}}\right)\left(\frac{1}{\alpha_1}v^{\frac{2}{\alpha_1}-1} \, e^{-\beta\, v^{\frac{1}{\alpha_1}} }+ ...
0
votes
1answer
66 views

My integral is behaving strangely

The following integral is something I am trying to solve for $$ \int_{\gamma}^{\infty} \Bigg[ 1- \left( \frac{2a(1+s x^{-1})+b}{1+s x^{-1} } \right) \Bigg] x^{\frac{2}{\alpha}-1} \, dx$$ We have ...
3
votes
1answer
38 views

Volume of a sphere with two cylindrical holes.

Consider a sphere of radius $a$ with 2 cylindrical holes of radius $b<a$ drilled such that both pass through the center of the sphere and are orthogonal to one another. What is the volume of the ...
1
vote
0answers
14 views

Compute the asymptotic expansion of the integral by Watson's Lemma

Use Watson's Lemma to find the asymptotic expansion of the following integral as $\lambda \to \infty$ with $\lambda>0.$ Assuming that $\phi (t)$ is infinitely differentiable on $[0,1].$ ...
0
votes
0answers
38 views

Integration: Step in paper unclear

I've seen in a paper the following step: $$2\operatorname{Re}\int_{\mathbb{R}^n} r \partial_r \bar u \Delta u \, dx=(n-2)\int_{\mathbb R^n} |\nabla u|^2$$ This is not clear to me as I calculated: ...
2
votes
1answer
23 views

Two integrals with bounds

I got an integral of the form: $\int_0^\infty da(\int_c^a db f(b))$ Is it somehow possbile to make it possible to integrate first over a? If yes, how?
1
vote
2answers
79 views

How to integrate $((x^2-1)(x+1))^{-2/3}$ using the substitution $u=(x-1)/(x+1)$?

I was asked to find the indefinite integral $$\int \frac{1}{((x^2-1)(x+1))^{2/3}} dx$$ using the substitution of $u=(x-1)/(x+1)$. How do I make this substitution? I attempted to solve this ...
0
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0answers
26 views

Do you obtain the same integral after this change of variable (solving of integral not needed)

Given the following integral $$T=\int_{\gamma}\left(1- c_1 - \dfrac{c_2}{1+s \, a \, v^{-1}}\right) v^{\frac{2}{\alpha}-1} dv $$ Let us do a simple transformation $$ v= s t^{\alpha}\rightarrow dv= ...
1
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1answer
26 views

Double integral of $e^{3+y^2}$ over a triangle

Evaluate $\iint_{A}^{} e^{3+y^2}dxdy$ where $A$ is a triangle with vertices $(0,0)$, $(0,-1)$ and $(1,-1)$. I don't know how to bite that. I tried multiplying it by $e^{x^2}$ and then changing the ...
2
votes
2answers
31 views

Does these sequence and series converge?

Let $f\in C^1[-\pi,\pi]$ st $f(-\pi)=f(\pi)$ and define $$a_n=\int^{\pi}_{-\pi} f(t)\cos nt dt\,$$ for $n \in\Bbb{N}$ . Then does the sequence $\{na_n\}$ converges? And does the series ...
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0answers
20 views

Help me understand procedure of integrals. [on hold]

Help me to calculate: $\int \sin (-x^2 )dx$ approximation by Taylor 2. series for $x_0=0$. Thank you
0
votes
2answers
19 views

How to find the corresponding matrix of a dot product over a polynomial ring to a specific basis

Let $V= \mathbb R[x]_{\leq 2}$ be the vector-space of real polynomials with degree $\leq 2$. We define a dot product on the $V$ as follows: $$\left<f,g \right> = \int_{0}^1f(x)g(x)dx.$$ ...
0
votes
3answers
30 views

How does integrating over absolute values work with definite integrals?

I have $ \int_0^\pi | \sin(x/2) | \, dx $, and according to Wolfram Alpha, the indefinite integral is: $$ -2\cos(x/2)\operatorname{sgn}(\sin(x/2)) + C $$ but the definite integral above evaluates to ...
0
votes
1answer
23 views

estimates Gaussian moments

Let $X_i \sim N(0,\sigma_i^2)$. Let $k\geq0$ be a fixed integer. I would like to compute $$A:=E[|X_1-X_2|^k|X_2|^k]$$ My idea was \begin{align*} A=&\int_{\mathbb{R}^2}|x_1-x_2|^k |x_2|^k ...
0
votes
1answer
35 views

$\frac{1}{2} \int_{a}^{b} f = \int_{a}^{c} f$ [on hold]

$f$ an integratable function defined in $[a;b] \rightarrow \mathbb{R}$: prove that exists $c \in [a;b]$ that: $\frac{1}{2} \int_{a}^{b} f = \int_{a}^{c} f$ and then give an example that might not ...
0
votes
1answer
29 views

When does the improper integral converge?

For positive numbers $p$ and $q$, find the condition for $p$ and $q$ such that the integral $$\int_0^{+\infty}\frac{dx}{x^p(1+x)^q}$$ converge. $x^p < (1+x)^p \Rightarrow x^p (1+x)^q < ...
1
vote
1answer
42 views

How to evaluate $\lim_{n\to\infty}\int_0^n\frac{x^2+a^2}{x^4+b^2x^2+b^4}dx$

Evaluate this limit: $$\lim_{n\to\infty}\int_0^n\dfrac{x^2+a^2}{x^4+b^2x^2+b^4}dx$$ I tried to simplify this fraction. I noticed that $x^4+b^2x^2+b^4$ can be written as $$\dfrac{x^6-b^6}{x^2-b^2}$$ ...
0
votes
2answers
20 views

Hyperbola question

the graph $ y^2=16x $ is a hyperbola; it can be rewritten as $ y= \pm 4\sqrt{x}$ when I draw it down however It is clearly not a function..question is whether it has to be one in order to perform ...
0
votes
0answers
15 views

Convergence of quadrature formulas and interpolating polynomials

There is a theorem of Polya (1933), which says: 1) If a interpolatory quadrature formula converges for all continuous functions on [a, b] and quadrature weights are all positive, then the formula ...
1
vote
1answer
22 views

A level Integration question.

1a) Prove that $$e^x\operatorname{sech} x\equiv\frac{2e^{2x}}{e^{2x}+1}$$ b) Find $$\frac{d}{dx}[\arcsin(\tanh x)]$$ Simplify your answer as far as possible. c) Hence, or otherwise, solve $$\int ...
2
votes
2answers
33 views

Integration by parts and $dx$ notation

Please overview this integral evaluation: $$ \int x^3 \arctan(x^2)dx = \frac{x^4}{4}\arctan(x^2) - \int \frac{1}{1+x^4}2x dx $$ Let's evaluate the right term: $$\int \frac{1}{1+x^4}\color{Blue}{2x ...
13
votes
3answers
307 views

How can we think and/or write rigorously about integration by substitution?

Define a function $I:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ as follows. $$I(a,b)=\int_a^b \sin t \cos t \,d t$$ Then we can find a more explicit description of $I$ using integration by ...
-1
votes
2answers
32 views

How to integrate $\int\frac{-x-1}{(x^2-2x+5)}dx$

How do you integrate $$\int\dfrac{-x-1}{(x^2-2x+5)}dx$$ ? I would be really grateful for an answer.
3
votes
2answers
112 views

Find $\int_0^a{f(x)}\, dx$

SMT 2013 Calculus #8: The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, ...
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votes
0answers
15 views

Need help to solve double integral exercise

I'm facing problems solving these integrals. I can't reach the result. Could anyone help me? There are two problems with the same integral. Integral: $\iint (y) dx dy $, a) $\{B=(x,y) \in R² | ...
0
votes
3answers
41 views

integrals calculation got wrong with the extra 2

Given $$ f(x, y) = \begin{cases} 2e^{-(x+2y)}, & x>0, y>0 \\ 0, &otherwise \end{cases}$$ For $ D: 0 <x \le 1, 0 <y \le2$, I'm trying to calculate this $$ \iint_D f(x,y) \, dxdy ...
0
votes
1answer
7 views

what is the value of $\int_{\lambda}(n_1(x,y)x+n_2(x,y)y)ds.$

If $n=(n_1(x,y)+n_2(x,y))$ is the outward unit normal at the point $P=(x,y)$ lying on the curve $\lambda$ which is $x^2+4y^2=4$, Then what is the value of ...
0
votes
3answers
47 views

Help with integration of $\frac{f'(x)}{[f(x)]^n}$.

How do I integrate an expression of the form $$ \frac{f'(x)}{[f(x)]^n} $$ with respect to $x$? Could I use some kind of recognition method, thus avoiding partial fractions? For example: $$ ...
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votes
0answers
14 views

Simplification of integral region (no integration skills needed)

We have the following "formula" or simplification for integrals: Let $f_i:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for $i=1,\dots,n$ and $g_j:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for ...
0
votes
0answers
36 views

Tricky multiple choice question about the properties of $\int_0^x\cos x\ e^{kx^4}\ \mathrm dt$

The question Let $$F(x) = \int_0^x\cos x\ e^{kx^4}\ \mathrm dt,$$ with $k \in \mathbb{Z}$. Which of the following is FALSE? a) $F$ does not have a horizontal asymptote for $k = 3$. b) ...
0
votes
1answer
48 views

$\int_0^{\pi/2}\ln(\sin(x))$?

From this paper: http://math.ucsd.edu/~ebender/20B/7_DefInt.pdf Shouldn't $du$ be $dt$? And also how do you get from that line to the final result if $du$ is not $dt$?
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votes
0answers
54 views

Help with calculus. what does a subscript k represent in (1)

I would like some help solving these problems. In (1) the maximum can be found by derivation right? where the derivative = 0, but what does " find the maximum a(subscript k)" mean? i keep seing it ...
0
votes
0answers
19 views

Existence of a potential function

Given some function $v\in C^1(\mathbb{R}^3,\mathbb{R}^3)$, how can we check whether there exists a function $f\in C^2(\mathbb{R}^3, \mathbb{R})$ such that $v = \nabla f$? I believe the existence of ...
2
votes
2answers
64 views

Evaluating $\int_0^\infty \frac{1}{(k-1)!} (\frac{x}{y})^{k+1} (1-y)^{-x/y} \, dx$

EDIT: I CHANGED THE QUESTION (I HAD THE WRONG BOUNDS!) THE ACTUAL QUESTION WAS FROM 0 TO INFINITY, NOT 0 TO 1! I'm stuck with evaluating this integral and I need some help! $$\large\int_0^\infty ...
0
votes
1answer
43 views

Any smart way of simplifying this integral ( example given before)

Assume we have non-negative variables $\alpha,\gamma,s$. The following integral $$ \int_{\gamma}^{\infty}\left( 1 -\frac{1}{1+s\, v^{-1}}\right) \,v^{\frac{2}{\alpha}-1}\, dv = \alpha \, F(\gamma \, ...
1
vote
1answer
29 views

Is this equality true

Is the following equality true? $$\int_{0}^{+\infty} \frac{x+1}{3x+4} = \int_{0}^{+\infty} \frac{x}{3x+4}+\int_{0}^{+\infty} \frac{1}{3x+4}$$
7
votes
2answers
110 views

Any advice on simplifying this nasty integral

Can anyone think of any smart way of approximating this nasty integral $$F = \int_{-c}^c f_X(x) \, dx $$ where $c$ is a non-negative constant (for example $\frac{1}{64}$) and where the integrand is ...