Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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Fix a parameter $h>0$ the following integral bounded by $C>0$?

Let $0<\alpha<1$, $h > 0$ and consider the following integral $$\int_1^\infty r^{-\alpha } e^{-hr} dr$$ Is this integral bounded by a constant $C>0$ independent of $h >0$? i.e $$\int_1^\...
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Continuity and integrability of a function $f:[0,1] \to l_2$

Good morning I’m working with this problem at the moment: Let’s say that we have a function $f:[0,1] \to l_2$ where base is $(e_i)=(0,0,0,….,0,1,0,…,0,0)$ and then $f(q_i)=e_i$ and $f(r)=Q$ we have to ...
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Is this inequality related to this integral correct?

Let $\alpha >0, 1/3 >\beta>0, T>0$, $h \in [0,T]$. In [Cerrai, Sandra. "Normal deviations from the averaged motion for some reaction–diffusion equations with fast oscillating ...
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General formula for logarithmic cosine integral

I am trying to find a general expression for $$ \int_0^{\pi/2} (\ln(\cos(x)))^n dx $$ for integer $n$, but have not been able to find it online or derive it. I have a good idea that the general form ...
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Any idea how to solve this definite integral? [duplicate]

i´m solving a problem, and this integral pops out, any idea for a substitution? $\int_{0}^{2\pi} \frac{cos^2x}{(1+\epsilon sinx)^2}dx$ where $0<\epsilon<1$ Thanks in advance
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2 votes
2 answers
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Interesting integral with fractional part function.

Question: Find $$\int_1^\infty \frac{\{x\}}{x(x+1)}dx,$$ where $\{x\}$ means $x - \lfloor x \rfloor$. I have attempted to split this into two integrals, namely $$\int_1^\infty \frac{x}{x(x+1)} - \...
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A Definite Integral under Absolute Value

I want to evaluate the following $$ \left|\operatorname{sgn}(a)\cdot\int_{-\frac{\sqrt{4ac-2}}{2a}}^{\frac{\sqrt{4ac-2}}{2a}}\left(c-\operatorname{sgn}(a)\sqrt{\frac{4ac-1}{4a^2}-x^2}-ax^2\right)dx\...
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1 answer
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Closed form of $\int_{0}^{\infty} \frac{x^{a} \ln^n x}{1+x^{b}} d x$ for natural number $n$?

I am going to evaluate the integral $$\displaystyle I=\int_{0}^{\infty} \frac{x^{a} \ln x}{1+x^{b}} d x, \tag*{} $$ where $b>a+1>0$, by its partner integral $$\displaystyle J(a)=\int_{0}^{\infty}...
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Evaluate the improper integral $\int_0^\infty x^{-1/2}e^{-(x-a)^2}dx$

Here is an integral I encountered in the wild: $$\int_0^\infty x^{-1/2}e^{-(x-a)^2}dx .$$ If we substitute $t = x^{1/2}, dt = \frac{1}{2}x^{-1/2}dx$, we have $$\int_0^\infty 2e^{-(t^2-a)^2}dt .$$ ...
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how to find the table integral that contains factorial of n?

I read in this page that" There are lots of definite integrals that depend on a parameter n∈N and whose result contains factorials of n " I am really struggle with this integral \begin{align*...
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Calculate the integral by Riemann $\int_{0}^{1}\sqrt{x}\,dx$

Calculate the integral by Riemann $\displaystyle \int _{0}^{1}\sqrt{x} \, dx$ $$\displaystyle a_{k} =0+k\cdot \frac{1}{n} =\frac{k}{n}$$ $$\displaystyle \Delta x=\frac{1-0}{n} =\frac{1}{n}$$ \begin{...
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Stieltjes integral and Riesz representation theorem

Would you please explain how I can derive eq(1.2) from eq (1.1) by using Stieltjes integral in the Riesz representation theorem?? And is it necessary to suppose that the time interval begins from -$\...
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computation of gaussian perimeter and volume

Let $B(0,1)$ denote the unit ball in $\mathbb{R}^n$ centered at the origin. I would like to compute the following weighted gaussian volume $$V_g(B)=\int_B \frac{1}{(2\pi)^{n/2}}\exp(-|x|^2/2) dx$$ and ...
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Compute $\displaystyle\int_0^{\pi}\dfrac{{\rm d}{\theta}} {\left({\epsilon}\cos\left({\theta}\right)+1\right)^2}$ [closed]

$$\int_0^{\pi}\dfrac{{\rm d}{\theta}} {\left({\epsilon}\cos\left({\theta}\right)+1\right)^2}$$
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Integral of $\{x\}$ from $e$ to $\pi$ [closed]

The question is: $$\int_e^\pi \{x\}\,dx$$ I have no idea where to get started, so any help is appreciated!
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Integral diverges but method works?

Based on this: https://math.stackexchange.com/a/4454826/595084 I want to find the integral $$I_1=\int^{\infty}_0\frac{\sin(x)}{e^x+1}\text{ d}x$$ In the method the answer uses, it converts this ...
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What are the steps to get that value integrating the given function?

How to calculate this integral $W=\int_0^{2\pi}\dfrac{6{\epsilon}{\mu}{\omega}{(R/C)^2}\cdot\left({\epsilon}\cos\left({\theta}\right)+2\right)\sin\left({\theta}\right)}{\left({\epsilon}^2+2\right)\...
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Understanding the way of integrating the below function by substitution

I understand from this post https://math.stackexchange.com/a/470460/961436 regarding why not injective is the required criteria . Consider a case of integral to be of form $\int_{0}^{1} \frac{1}{1-x+x^...
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Three complicated integrals lead to an anodyne expression

Someone being forgotten by me has claimed: $$\int_{0}^{1} \frac{1}{(1+x^2)\sqrt{1-x^2}\sqrt{4-x^4}\sqrt{9-x^4} } \text{d} x -\int_{0}^{1} \frac{1}{(3+x^2)\sqrt{1-x^4} \sqrt{2+x^2}\sqrt{4+x^2}\sqrt{5+...
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Prove that $\int_{0}^{1} x K(x) E(x) \,\mathrm{d} x{=}\frac{9}{16}+\frac{21}{32} \zeta(3)$

Background let $\displaystyle I_k=\int_{0}^{1} x^k K(x) E(x) \,\mathrm{d} x$. where $K(z)$ is the complete elliptic integral of the 1st kind, $E(z)$ is the complete elliptic integral of the 2nd kind. ...
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How to calculate functions that are being integrated?

I would like to create an estimate for the energy dissipated in a MOSFET transistor during on switching. The off state, the on state as well as the transition time are known. I know the related ...
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Is it true $\sum\limits_{i=1}^n \sum\limits_{j=n-i+1}^n i^{-a}j^{-a} \ge \int_1^n x^{-a}(\int_{n-x+1}^n y^{-a}\, \mathrm{d}y)\, \mathrm{d} x$?

Conjecture 1: For each $n\ge 1$ and $a \ge 2$, $$\sum_{i=1}^n \sum_{j=n-i+1}^n i^{-a}j^{-a} \ge \int_1^n x^{-a}\left(\int_{n-x+1}^n y^{-a}\, \mathrm{d}y\right) \mathrm{d} x.$$ This is related to this ...
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how should I the double integral, not really consistent with my understanding

I have no trouble to those integral except this one $$ \iint_Rf(x,y)\,dA=V=\int^b_a\int^d_cf(x,y) \,dy\,dx\quad\text{R here indicate a region}$$ I agree with $$A(x)=\int^d_c f(x,y)\,dy=\int^d_c\,dA(x)$...
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Integrating $\int_{1}^{\alpha} x^n(x^2-1)^{q-\frac{1}{2}}dx$

Let $\alpha>1$, $n \in \mathbb{N}$ and $q\geq0$. Which methods is possible to use to solve this integral? $$\int_{1}^{\alpha} x^n(x^2-1)^{q-\frac{1}{2}}dx$$ I tried using the computer for especific ...
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1 answer
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Taking an Integral with respect to time to find the work done on a rocket.

I was given a rocket problem for my Calculus 2 class and want to know if it's possible to take an integral of time to find the correct solution. I think it will be easier to explain if I show the ...
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Evaluate $\lim \limits_{n \to \infty } n\cdot a_n\cdot a_{n+1}$ where $a_n =\int_0^{\pi/2}((\sin x)^n) dx$

The question is to evaluate the limit $$\lim \limits_{n \to \infty } n\cdot a_n\cdot a_{n+1}$$ where $$a_n =\int_0^{\pi/2}((\sin x)^n) dx$$ I just evaluated the limits for natural numbers $n$ and got $...
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2 votes
1 answer
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Solution verification of an approximation of an integral

I have to show that \begin{align} \int_{0}^{1} e^{-xt} \sin t dt \approx \dfrac{1}{x^2}. \end{align} It is well known that the above integral belongs to as subtype known as Laplace Integrals and to ...
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1 vote
1 answer
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Definite Integral of $\int_{-L}^{L} \cos(\frac{nπx}L)\cos(\frac{mπx}L)dx$

Please can someone explain why I need to substitute the values of integers $m$ and $n$ in the definite integral: $$I=\int_{-L}^{L} \cos\left(\frac{nπx}L\right)\cos\left(\frac{mπx}L\right)dx = \begin{...
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Triple integral set up using cylindrical coordinates

Set up an integral in cylindrical coordinates to evaluate $\iiint_{E} x y d V$ where $E$ is the region enclosed by the cone $z=2-\sqrt{x^{2}+y^{2}}$, the cylinder $x^{2}+y^{2}=1$, and the $x y$ plane. ...
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3 answers
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Evaluate the integral $ \int_0^{ + \infty } {\frac{{x^2 - 1}}{{x^2 + 1}}\frac{{\sin x}}{x}dx} $

So I have an assignement to do and it has multiple integrals, I did all of them but this one I can't seem to know how to do it. $$ \int_0^{ + \infty } {\frac{{x^2 - 1}}{{x^2 + 1}}\frac{{\sin x}}{x}...
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2 votes
1 answer
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Showing $\int _{0} ^{\pi/4} \frac{\cos^{2022}(x)}{\sin^{2022}(x) + \cos^{2022}(x) } dx \approx \frac{\pi}{4}$

Show that $$\int_{0} ^{\pi/4} \frac{\cos^{2022}(x)}{\sin^{2022}(x) + \cos^{2022}(x) } dx \approx \frac{\pi}{4}$$ My method was this: I tried using $x \to \pi/4-x$ conversion but that doesn't lead to ...
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3 votes
0 answers
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Understanding when we need injectivity requirement for substitution in integrals and when not and why

I am expanding my query i asked before so as to get a more well explained reasoning for this :. Suppose we are required to evaluate three integrals : $\int_{0}^{4} \frac{2x-4}{x^2 - 4x + 5}dx$ $\...
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1 answer
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Understanding why even though derivative exists but the derivative is not integrable in the given x range

Why is the derivative which exists everywhere for $f(x) = x^2 \sin(\frac{1}{x^2})$ for $x \neq 0$ and $f(x) = 0$ for $x=0$ is not integrable over a region $[a,b ] \in \mathbb{R}$ having $0$ included ...
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2 votes
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Expressing $\int_0^\pi \sin^n(x)dx$ in terms of the gamma function

Let $I_n = \int_0^\pi \sin^n(x)dx$ and suppose that we have already established a recursive relation $I_n = \frac{n - 1}{n}I_{n-2}$ and we know that $\Gamma(x + 1) = x\Gamma(x), \Gamma(1/2) = \sqrt{\...
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8 votes
4 answers
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How to evaluate $\int^{\infty}_0 \frac{x^{1010}}{(1 + x)^{2022}} dx$?

How to evaluate the following integral? $$\int^{\infty}_0 \frac{x^{1010}}{(1 + x)^{2022}} dx$$ Here's my work: $$I = \int_0^\infty \dfrac{x^{1010}}{(1+x)^{2022}} dx = \int_0^\infty \dfrac{1}{x^{1012}(...
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1 answer
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Checking the approach for solving the integral

I am trying to show that $\int_{a}^{b} \frac{f'(x)}{f^2(x)} dx = \frac{f(b) -f(a)} {f(a) f(b)}$ given that $f(x)$ is, continuous in $[a,b]$, differentiable in $(a,b)$ and $f(x) \neq 0$ in $[a,b]$. New ...
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0 answers
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Integrating $3 \cos^2 x \sin x$ [closed]

$$I = \int_0^1 3 \cos^2 x \sin x \,{\rm d}x$$ Please show it step by step and explain each step.
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Relation between normalized product of integrals of two functions and integral of product of two normalized functions

Could someone let me know if there is any relation between A and B, if $A=\frac{\int^L_0 f(x)g(x)dx}{L}$ and $B=\frac{\int^L_0 f(x)dx\cdot\int^L_0 g(x)dx}{L^2}$
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Prove that the integral $\int _0^{\frac{\pi }{2}}\:\frac{1}{\sin\left(x\right)+\cos\left(x\right)}dx$ is bounded [closed]

I need to prove the following claim without solving the integral, can anyone help me with this? $$\frac{\pi }{2\sqrt{2}}\le \int _0^{\frac{\pi }{2}}\:\frac{1}{\sin\left(x\right)+\cos\left(x\right)}dx\...
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2 answers
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How to interpret integrals that have conditions written beside them

sorry if this question has been asked before. I tried finding similar questions but couldn't find any. I have very little background in statistical mechanics, but I have been reading some literature, ...
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Find the value of 'K' .

Let $\alpha , \beta \in R$ such that $2(\alpha-1)^3+3(\alpha-1)^2+6\alpha=0$ and $2(1-\beta)^3+3(1-\beta)^2-6\beta+12=0 $ and $\alpha \neq \beta$ , also graph of $y=f(x)$ is symmetric about the point $...
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2 votes
1 answer
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Confused what I did wrong for $\int_{0}^{\infty} \frac{1}{1 + x^4} \, dx$

I did $ x = u\sqrt{i}$ $$\sqrt{i}\int_{0}^{\infty} \frac{1}{1 - u^4} \, du$$ $$\sqrt{i}\int_{0}^{\infty} \frac{1}{1 - u^2} \cdot \frac{1}{1 + u^2} \, du$$ $ v = \tan^{-1}(u)$,$dv = \frac{1}{1 + u^2} ...
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0 votes
1 answer
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Indication: $\displaystyle \lim_{x \to \infty}{\int^{x}_{1}{\ln(f(t))dt}}$

Let $f$ be a function $f:\mathbb{R}\to\mathbb{R}$, $f(x)=4x^3 + 1$. I have to find out $\displaystyle \lim_{x \to \infty}{\int^{x}_{1}{\ln(f(t))dt}}$. My intuition is telling me the limit is $+\infty$....
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1 answer
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Multiplying definite Integrals [closed]

$\int_a^b f(x)\,dx \cdot \int_a^b g(x)\,dx$ I would assume it would equal to $\int_a^b f(x)\cdot g(x)\,dx$ is this right? is there another solution?
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1 answer
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Gaussian integral in 3rd dimensions

I have been wondering about computing $(1/3)!$ and using the Gamma function. After substituting for $x=t^{\frac{1}{3}}$, I got $\int_{0}^{\infty}e^{-x^{3}}dx$. May I know if there is any way to ...
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0 answers
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Columbia Integration Bee 2022 Finals [duplicate]

I want to find the definite integral shown below, but I'm not quite sure where to start. The fastest solution apparently involved some sort of change of variables, but I can't quite find a ...
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1 vote
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$y=3x$ divides the region between $y=1-x^2$ and $y=x^2-1$ into two parts. What is the area of the smaller region?

$y=3x$ divides the area enclosed by $y=1-x^2$ and $y=x^2-1$ into two parts. What is the area of the smaller region? First I calculated the area between $y=1-x^2$ and $y=x^2-1$, $$S=\int_{-1}^1(1-x^2)...
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-1 votes
3 answers
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How to show that $\int_{0}^{\infty}\frac{x^{n-1}}{1+x}dx=\frac{\pi}{\sin{n\pi}},$ where $0<n<1$ [duplicate]

$$\int_{0}^{\infty}\dfrac{x^{n-1}}{1+x}dx=\dfrac{\pi}{\sin{n\pi}},\text{ where }0<n<1$$ I want to get the result on right. I substituted $x^{n-1}=t$ but when forwarded to transform in $t.$ The ...
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-1 votes
0 answers
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Calculating probability with a given pdf [closed]

Please check out the image attached. I have tried integrating the function from 0.3 to 1 but I am unable to remove the delta term.
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2 votes
2 answers
53 views

Consider function $y=f(x)$ whose derivative's domain is $\mathbb R$ and $xf'(x)=e^x-1, f(1)=0$. Determine the value of $\int_0^1xf(x)\,\mathrm dx$.

Consider function $y = f(x)$ whose derivative's domain is $\mathbb R$ and $xf'(x) = e^x - 1, \forall x \in \mathbb R, f(1) = 0$. Determine the value of $\displaystyle \int_0^1xf(x)\, \mathrm dx$. [...
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