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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Problem about limit of an integral

I came across this question while doing some exercises on integrals, and I was wondering if I could get some help. a) Show that for $n < -1$, $\int_1^N x^n dx$ converges as $N \to\infty$, and for ...
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1answer
34 views

Estimating the integral $\int \frac{\sin(x)}{x}\, dx$. [on hold]

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 ...
1
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1answer
28 views

Find the derivative of an integral.

Find the derivative of the following integral $$ F(x)=\int_x^{x^2}e^{t^7}dt $$ Find F′(x) given F(x). Normally I would show my attempt in working out the problem: however, I don't even know where ...
2
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1answer
83 views

Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
1
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1answer
43 views

How to integrate $1/\sqrt{(1+x^2)^3}$?

Normally I use WolframAlpha pro to help me with problems I don't know however wolfram wont/cant show me the steps only the final solution to this integration problem. Is anyone able to assist me with ...
1
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1answer
29 views

Simple Trig Integration. Why is my answer wrong?

$$\int \dfrac{\cos x+\sin 2x}{\sin x}dx=\int \dfrac{\cos x+2\cos x\sin x}{\sin x}dx=\int \dfrac{\cos x\left(1+2\sin x\right)}{\sin x}dx$$ Substitute $u=\sin x$ and $du=\cos x\ dx$: ...
3
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1answer
44 views

Riemann sum $\int x^m dx$?

I'm trying to find the Riemann integral of $x^m$ between $a$ and $b$ with $b>a$. So far I have managed to get $$\int_a^b x^m~dx=\lim_{n\rightarrow \infty}\left(a^m \times \frac{b-a}{n} \times ...
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0answers
20 views

Show that $\int_a^b f(x) dx=\lim_{n\rightarrow \infty} \sum_{k=0}^{n-1} \int_{x_k}^{x_{k+1}} f(x) dx$.

I've come up with a proof for the following statement, but I'm not quite sure it's 100% correct. I would appreciate any help: If $f$ is integrable on $[a,b]$, $x_0=a$, and $x_n$ is a sequence of ...
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0answers
18 views

How to evaluate $\int_{-\infty}^t v(\alpha)\,d\alpha$ for $v(\alpha) = \sin\alpha$?

Integral form of the inductor's V-I relation is: $$i(t) = \frac{1}{L} \cdot \int_{-\infty}^t v(\alpha)\,d\alpha$$ How can I determine this function for $v(\alpha) = \sin\alpha$? I've tried to but I ...
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0answers
25 views

The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles.

Find the volume of the following solid S: The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles. So far I got ...
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0answers
20 views

exponential integration with fractional powers

I am trying to solve the following integral $$\int_{-\infty}^a \frac{\beta_1 \beta_2}{y^2(c-y)^2} e^{-\beta_1/(c-y)} e^{-\beta_2/y} \, dy$$ where $a<0$, $c>0$, $\beta_1>0$, $\beta_2>0$ I ...
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2answers
36 views

Study the convergence of $\int_1^\infty \frac{x\ln x}{x^4-1} dx$

Study the convergence of $\int_1^\infty \frac{x\ln x}{x^4-1} dx$ So first we have two potentially problematic points which are $1,\infty$ We split the integral to $$\int_1^2 \frac{x\ln x}{x^4-1} ...
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3answers
47 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?
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1answer
26 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 ...
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0answers
14 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) = ...
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1answer
31 views

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

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?
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1answer
29 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
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2answers
37 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... $$\int_0^{\ln(2)} \sqrt{(e^x-1)} dx= \int_0^{\ln(2)} (e^x-1)^{1/2} dx= \frac{2}{3}(e^x-1)^{3/2} |_0^{\ln(2)}$$
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0answers
24 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: ...
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0answers
33 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 ...
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2answers
71 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.
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0answers
29 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 $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
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0answers
54 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 ...
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2answers
101 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 ...
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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. ...
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2answers
111 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}} }+ ...
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1answer
71 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
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1answer
44 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 ...
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0answers
15 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 $[0,1].$ ...
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0answers
41 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
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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?
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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 ...
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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= ...
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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
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2answers
32 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
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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: $$\left<f,g \right> = \int_{0}^1f(x)g(x)dx.$$ ...
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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 ...
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1answer
25 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 ...
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1answer
36 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 ...
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1answer
30 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 < ...
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1answer
46 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 $$\dfrac{x^6-b^6}{x^2-b^2}$$ ...
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2answers
21 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 ...
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0answers
17 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
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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
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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
319 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
33 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
114 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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0answers
16 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 | ...