Mathematics Stack Exchange
Mathematics Stack Exchange

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.

20559 questions
1
vote
1 answer

How to calculate the integral $\int_0^2 { \int_0^{1/2x_1} {\frac{-1+x_1x_2-2x_2}{x_1-2x_2}} }dx_2dx_1$

This problem (if my derivations of them are correct) lead me to calculate the following integrals: $$I_1 = \int_0^2 { \int_0^{\frac{1}{2}x_1} {\frac{-1+x_1x_2-2x_2}{x_1-2x_2}} }dx_2dx_1$$ $$I_2 = \int_0^2 { \int_{\frac{1}{2}x_1}^1…
ploosu2
  • 8,707
1
vote
1 answer

Wrong integral proof

I was told by my maths teacher that the following procedure is wrong but didn't really understand why so I hope someone here can explain it to me. We want to prove that $\int_0^{π/2} f(sinx)dx=\int_0^{π/2} f(cosx)dx$ I thought of setting…
user92596
1
vote
1 answer

Iterated Integral

I look for an argument to this statement: $$ \int_a^x dx_1 \int_a^{x_1} K(x_1,t) dt= \int_a^xdt \int_t^x K(x_1,t) dx_1 $$ It is certainly an integration by change of variables that I can not clarify
Zbigniew
  • 805
1
vote
1 answer

Area of one turning of Archimedian spiral

So, if the Archimedian spiral is given with formula $r=2\theta$, what does that formula represent and what is the area of one turning of the spiral? The teacher solved it…
A6SE
  • 1,645
1
vote
1 answer

Rearranging double integral and bounds

I am trying to figure out why we can rewrite $\int_0^n s (\int_0^s 1 \, dt) \, ds = \frac{n^3}{3}$ as $\int_0^n 1 (\int_s^n t \, dt) \, ds = \frac{n^3}{3}$ I would appreciate any pushes in the right direction. I'm not quite sure where to start.
jason.tran
  • 27
1
vote
1 answer

Solution of Integral

I want to solve the following integral $$ \frac{\alpha \beta}{2} \int_0^\pi \cos\theta \sec^{2}\theta(\tan(\theta/2))^{-\beta-1} (1+\gamma(\tan(\theta/2))^{-\beta})^{-\frac{\alpha}{\gamma}-1}d\theta$$ after subtituting…
SAAN
  • 300
1
vote
2 answers

How to take this integral? It looks like as Gamma but I'm confused.

x and b are real. b is constant. $$\int _{ -\infty }^{ +\infty }{ { e }^{ { (ix+b) }^{ 2 } } } dx$$ What the answer should be?
Sina
  • 157
1
vote
2 answers

Integrate $\int_0^{\pi/2}\cos^8(x)dx$

I'm usually pretty good with definite integrals, but this one's got me completely lost. Any help is appreciated! (And the sooner the better, please!) $$\int_0^{\pi/2}\cos^{8}x\,dx.$$
Katy
  • 21
1
vote
0 answers

Gaussian integral solution $\int\limits_{-\infty}^t e^{-\frac{1}{2 a}(c-x)^2} dx$

I want to compute the following integral $$\int\limits_{-\infty}^t e^{-\frac{1}{2 a}(c-x)^2} dx$$ I substitute $y=\frac{c-x}{\sqrt{a}}$. Thus I have $dy=-\frac{dx}{\sqrt{a}}$ and the upper limit $\frac{c-t}{\sqrt{a}}$. Then I…
ana
  • 11
1
vote
1 answer

Calculation of an integral based on $\int^{\pi/2}_0 f( \sin x ) dx = \int^{\pi/2}_0 f(\cos x) dx$

Given a continuous function $f:[0,1] \to \mathbb {R}$ Prove that $$\int^{\pi/2}_0 f(\cos x)dx = \int^{\pi/2}_0 f(\sin x) dx$$ Calculate $$I= \int^{\pi/2}_0 \frac{\sin^2 x + \sin x}{1 + \sin x + \cos x} dx$$ The 1st was solved by letting $g:[0,1]…
bolzano
  • 1,032
  • 6
  • 17
1
vote
2 answers

Evaluate the definite-integrals $\int_0^{\pi} \frac{\sin nx}{\sin x}dx$

Evaluate the definite-integrals $\int_0^{\pi} \dfrac{\sin nx}{\sin x}dx$ my teacher say that, using the formula : $\sin nt=\dfrac{e^{ni}-e^{-ni}}{2i}$, but i can't :(.
Road Human
  • 1,012
1
vote
0 answers

Calculate the arclength of $y=\frac{\sqrt{x}(x^2+2)}{3}, \hspace{0.5cm} 2\leq x\leq 4$

Calculate the arclength of $$y=\frac{\sqrt{x}(x^2+2)}{3}, \hspace{0.5cm} 2\leq x\leq 4$$ My work: I have calculated $$1+(\frac{dy}{dx})^2=\frac{25x^4+20x^2+36x+4}{36x}$$ I need your help to solve the integral…
Flip
  • 818
1
vote
2 answers

How to prove $L_{f}(P) \leq L_{f}(Q)$ when $Q$ and $P$ are partitions of $[a,b]$ and $Q \supseteq P$

I'm having trouble proving this idea. Suppose that $f$ is bounded on the interval $[a, b]$. $P$ and $Q$ are partitions of $[a, b]$, and $Q \supseteq P$. $$ L_{f}(P) \leq L_{f}(Q) $$ I know that this makes sense, because by adding points to a…
Backslash
  • 345
1
vote
1 answer

Integral $\int_0^\infty \frac{e^{-cy} dy }{1+ay}$

$$I'=\int_0^\infty \frac{e^{-cy} dy }{1+ay}$$ a, b, and c, are positive coefficients. This integral is part of a problem which I'm trying to solve it and after lot's of effort the problem transform into two parts. For solving the second part(the…
erfan taherkhani
  • 55
1
vote
0 answers

evaluating an indefinite and improper integral

$$ I=\int_0^{+\infty} \frac{Q_1(a,\sqrt{by})}{1+cy}dy $$ $a$, $b$, and $c$ are positive coefficients. $Q_1$ is Marcum $q$-function. This integral is part of a problem which I'm trying to solve it and after lot's of effort the problem transform into…
erfan taherkhani
  • 55
1 2 3
…
76 77