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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

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Can $\int_{0}^{\infty}\frac {\cos{x}}{(1 + x^2)} dx$ be evaluated without complex analysis?

Can the integral $\int_{0}^{\infty}\frac {\cos{x}}{(1 + x^2)} dx$ be evaluated by differentiation under integral or any other method without involving complex analysis? I tried using the function $\cos{x}\exp(-m(1+x^2))/(1 + x^2)$ Then used $\cos{x}…
Archisman Panigrahi
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Definite integral of a bounded function on a bounded interval depends at least linearly on integration extremes?

I think that the title of the thread summarises all I would like to know about an integral of one variable only. The question may appear silly (maybe it actually is), but it stems from the idea of the raw approximation of an integral provided by…
Bounded
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hard indefinite integral with unreal solution

Can someone give me an idea on how I should solve this? $$\int \frac{1}{2 \sin{\left (x \right )} + 5 \cos{\left (x \right )}}\, dx$$ I tried to enter a replacement, but I'm not sure how to continue. Here's the exercise. result
TaumCkBep
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Double integral over region bounded by nonparallel lines

Problem I'm having trouble with the following integral: $\int \int_D \frac{x}{y} dx dy $ where D is the area bounded by $1 \leq 2x+y \leq 5, \quad 4x \leq y \leq 8x$ My attempt at a solution Substitution: $ u = 2x + y, \quad v= y \ \implies x =…
Mattias Johnson
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Calculating area between two curves

See this link to get a picture of what I mean. If you want to calculate the area between $f(x)$ and $g(x)$ on a certain interval $[a,b]$. Do I have to add the two areas between the $x$-axis and $f(x)$, the $x$-axis and $g(x)$, or can I just find the…
user527567
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What is $\int_0^\infty \int_x^\infty \frac{1}{y} e^{-y/2} \, dy \, dx$?

Frankly I don't know how to simplify or which theorem to apply. I do know theorem on $e^{-y^2}$ (gaussian) but I don't think I can manipulate the current question to that form? P.S Answer is 2.
ManishKumar Singh
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Range of $S,T$ in $2$ sums

If $\displaystyle S = \sum^{4n-1}_{k=3n}\bigg(\frac{k^2-7kn+13n^2}{n^3}\bigg)$ and $\displaystyle T= \sum^{4n}_{k=3n+1}\bigg(\frac{k^2-7kn+13n^2}{n^3}\bigg)$. then which one is/are right $\; (a)\displaystyle \; S<\frac{5}{6}\; (b)\;…
DXT
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Definite integral of $\sin x\cos x$ from $0$ to $2\pi$?

I've done this in two different ways. First: $$\int _0 ^{2 \pi} \sin x \cos x dx = 2 \int _0 ^\pi \sin x \cos x dx = 0,$$ since $\cos(\pi - x) = - \cos x$. By using the property $$\int _0 ^{2a} f(x) dx = 0 \text{ if } f(2a-x)=-f(x)$$ or $$\int…
Flame Striker
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Finding $\sum^{\infty}_{n=1}\int^{2(n+1)\pi}_{2n\pi}\frac{x\sin x+\cos x}{x^2}$

Finding value of $\displaystyle \sum^{\infty}_{n=1}\int^{2(n+1)\pi}_{2n\pi}\frac{x\sin x+\cos x}{x^2}$ Try:$$\frac{\cos x}{x} = -\bigg(\frac{x\sin x+\cos x}{x^2}\bigg)$$ So $$\sum^{\infty}_{n=1}\bigg(\frac{\cos…
DXT
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Properties for the revision of definite integral

I am revising the syllabus of mathematics and I think I have done enough properties of definite integration. I am listing all the properties that I have studied: $\lim_{n\rightarrow…
userNoOne
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Sommerfeld Integral Evaluation - Hankel Integral

I was studying theory of electromagnetic from the book Time-harmonic Electromagnetic Fields by Roger Harrington and what caught up my attention was following identity: $$ \frac{\mathrm{e}^{-\mathrm{i}k_{0}r}}{r}…
Milad Mokhtari
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Assume that $g:\mathbb{R}\rightarrow\mathbb{R}$ is function such that any integral of the form $\int_{-\infty}^{t}g(x)dx$ is finite ...

I want to know if the following statement is true. Assume that $g:\mathbb{R}\rightarrow\mathbb{R}$ is function such that any integral of the form $\int_{-\infty}^{t}g(x)dx$ is finite and there exists a limit…
Lukaszmat
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Definite Integral Ruining My Life

So I need to prove that any function that fulfills Equation 1 also fulfills Equation 2 for an arbitrary small value of $k.$ Equation 1: $f'(x)=k\frac{f(x)}{f(x)+1-k}$ Equation 2: $f(n+1)-f(n)=k\frac{f(n)}{f(n)+1-k}$ What I have tried: I have been…
Dying at an increasing rate
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Show $\int_0^{2\pi}\cos(n\phi')\cos^l(\phi-\phi')\mathrm{d}\phi=\frac{2\pi}{2^l}\cos(l\phi)\delta_{l,n}$

I have to show $$\int_0^{2\pi}\cos(n\phi')\cos^l(\phi-\phi')\mathrm{d}\phi=\frac{2\pi}{2^l}\cos(l\phi)\delta_{l,n}$$ where $l,n$ are positive integers such that $l\leq n$ I'm supposed to use the fact $$…
user438666
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Compute the double integral

I want to compute the following double integral: $$\int_0^1dx\int_x^1xe^{y^3}dy$$ I'm can't seem to get the right answer though.. Using integration by substitution ($u = y^3$, $du = 3y^2 dy$) I…
guest
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