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 ...

learn more… | top users | synonyms (3)

0
votes
0answers
6 views

Equation of a line tangent to g(x) and parallel to line connecting endpoints of g(x)

Let g(x) be a differentiable function defined on the interval 0 ≤ x ≤ 16. Some values of g(x) and its derivative g'(x) are given below. Which of the following is the x-intercept of the line tangent to ...
-1
votes
0answers
16 views

Find the value of : $\int_{\mathbb{R}\setminus\left[-1,1\right]}\hat{f}^{2}\left(\xi\right)\left(\xi^{2}-1\right)d\xi $

Assume that $\hat{f}\in L^2(\mathbb{R})$, how can we compute this integral $$\int_{\mathbb{R}\setminus\left[-1,1\right]}\hat{f}^{2}\left(\xi\right)\left(\xi^{2}-1\right)d\xi $$ where $\hat{f}$ is ...
0
votes
0answers
22 views

Evaluate $\frac{1}{2\pi}\int_{-\pi}^\pi g(x) dx$ where $g(x) = \int_{0}^x f(t) dt$

Let $f$ be a $2\pi$-periodic function such that $\int_{-\pi}^\pi f(t) dt = 0$. Define $g(x) = \int_{0}^x f(t) dt$. Evaluate $$\frac{1}{2\pi}\int_{-\pi}^\pi g(x) dx$$ I hope the integral is equal ...
2
votes
0answers
17 views

Double integrals to find area?

In my math textbook it's written that double integrals is used to find area. Is it true? I thought I'd used to find volume. If so, how is it used to find area?
2
votes
1answer
34 views

How to evaluate $\int_{0}^{2\pi}dx\!\int_{0}^{\pi}\sin ye^{\sin y(\cos x-\sin x)}dy=\sqrt{2}\big(e^{\sqrt{2}}-e^{-\sqrt{2}}\big)\pi $

How to eveluate this double integral below, $$\int_{0}^{2\pi}\mathrm dx\!\int_{0}^{\pi}\sin ye^{\sin y(\cos x-\sin x)}\mathrm dy=\sqrt{2}\big(e^{\sqrt{2}}-e^{-\sqrt{2}}\big)\pi $$ I only got an ...
0
votes
1answer
30 views

using integrals, how do I find the average distance between an origin and points on a circle $(x-2)^2+y^2=4$

The advice given is to use $ \int \sqrt{1-\cos x}$ twice, once from $ 0 $ to $\pi$ and then $\pi$ to $ 2\pi$. I don't completely understand how this is relevant to finding the average distance.
0
votes
1answer
34 views

solving the integral $\sqrt{\sin(y)}(4-y)$

So the question asked me to find the volume of the area between $\sqrt{\sin(y)}$ and $x=0$ rotated about $y=4$. I came up with the integral $$\int_0^{\pi}(4-y)\sqrt{\sin(y)}dy$$ I can't figure out ...
1
vote
4answers
62 views

$\int \frac{1}{\sqrt{x^2+1}} dx$

So I've seen some options on the internet that are fairly good, but I have this substitution: $x^2+1=t-x$, you square both sides and get $x = (t^2-1)/t$ and $x + 1 = (t^2-1)/2t + 1$. If we call that ...
1
vote
2answers
40 views

Integrating $\int\frac{1}{kx}dx$ [duplicate]

$\int\frac{1}{kx}dx$ There are two ways to integrate this... Method 1 (Separating the coefficient from the variable): $\frac{1}{k}\int\frac{1}{x}dx$ $\frac{\ln|x|}{k} + c$ Method 2 (knowing that ...
0
votes
1answer
37 views

Integrate $\frac{\sin x^3}{x^3}$ as a power series

Today, I tried to do this by taking the MacLaurin of Sin to 4 terms, putting in $x^3$ in place of $x$, multiplying the terms by $x^{-3}$, and integrating. I came out with a sum that had $x^{6n+1}$ as ...
0
votes
2answers
27 views

Complex integration problem via Cauchy's integral formula

I want to integrate the following :$$\int_{|z|=2} \frac{dz}{z^{2}-1}$$ in the positive direction. So my idea is two split the integral into a sum of two integral , something like $$\int_{|z|=2} ...
-1
votes
2answers
48 views

Solve defined integral exp(x^2+x) from -inf to inf [on hold]

Solve $$\int_{-\infty}^{\infty} e^{x^2+x} \,dx$$
0
votes
3answers
21 views

Revolve around x-axis

I have this equation, $f(x)=x(2\sin(x)+x\cos(x))$ that I need to revolve around the $x$-axis from $x=0$ to $x=2$. I found this integral of $f(x)$ to be $x^2\sin(x)+C$. I am looking at using the ...
1
vote
1answer
36 views

Does $\int f(u) \, du \int g(v) \, dv = \int \int f(u) g(v) \, du \, dv $?

I saw a proof of $\Gamma (\frac{1}{2})=\sqrt{\pi}$ that had the step $$\left[ \Gamma\left(\frac{1}{2}\right) \right]^2 = \left\{2 \int_0^\infty e^{-u^2} \, du\right\} \left\{ 2 \int_0^\infty ...
-1
votes
0answers
48 views

Show that, $\int_0^\infty\frac{1}{(1+x^a)^b}dx…$ [duplicate]

$(a,b)>1$ Show that, $$\int_0^\infty\frac{1}{(1+x^a)^b}dx=\frac{\pi}{a}\cdot\frac{1}{\sin\left(\frac{\pi}{k}\right)}\cdot\prod_{s=1}^{b-1}\left(\frac{as-1}{as}\right)^{(-1)^{s-1}}$$ If ...
-3
votes
0answers
27 views

If $f = g +h$ then $\int_E f = \int_E g + \int_E h$ is independent of the choices of $g$ and $h$ [on hold]

Let $f$ be a measurable function on $E$ which can be expressed as $f = g +h$ where $g$ is a finite and integrable function over $E$ and $h$ is nonnegative over $E.$ Define $\int_{E} f = \int_E g + ...
1
vote
1answer
29 views

Combining error terms in Simpson's rule

My numerical analysis textbook (Burden and Faires) derives Simpson's rule as $$\begin{align} \int_{x_0}^{x_2}f(x)\,dx&=2hf(x_1)+\frac{h^3}{3}f''(x_1)+\frac{h^5}{60}f^{(4)}(\xi_1) ...
0
votes
0answers
34 views

Approximating functions using Taylor polynomials

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide any explanations. I literally have no idea how to ...
4
votes
0answers
57 views

How to get to the closed form of $\int_{-\infty}^{\infty} \frac{x^2e^x}{(e^x+1)^2}$ [duplicate]

I came across this integral when helping some friends with a statistical mechanics assignment, Mathematica reports it as $\frac{\pi^2}{3}$. So far I have noticed that the integrand is an even ...
0
votes
1answer
31 views

Convert Integral Rectangular to Polar

How can convert this problem $$ \int_0^2 \int_x^\sqrt{8-x^2} \left(x^2+y^2\right)^{3/2} dydx $$ I convert limits and funtion to polar cordinates as follows: $$ \begin{split} r^2 &= ...
0
votes
2answers
32 views

Logistic differential equation problem

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide any explanations. I'm having trouble turning the ...
0
votes
1answer
45 views

If the integral of $|f_n|$ converges to zero, so does the integral of $f_ng$ for integrable $g$

Let's assume that the $f_n$ are integrable and all bounded with the same bound. also, that $\int|f_n|\to 0$ as $n\to\infty$ (a) Prove under these assumptions that $\int f_ng\to0$ for any $g\in ...
1
vote
5answers
51 views

Prove that $\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$

Is it possible to prove that $$\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$$ without evaluating the integral?
1
vote
0answers
26 views

Showing a function is defined

(a) Defined $f$ by $f(y):=\int_0^\infty\frac{xy}{(x^4+y^4)^{3/4}}dx$. Prove $f(y)$ is defined (i.e integral exists) for every $y\in\mathbb{R}$. (b)Prove that actually $f(y)=c\space sign \space y$ for ...
1
vote
1answer
20 views

Integrals with functions as bounds

How to calculate integral such as $$\int_{g(χ)}^{φ(χ)} f(s) \, ds$$ where $F'(s)=f(s)$ Integrals like this appear often in PDE's .I'd like to know the whole theory i mean if there is a formula how ...
3
votes
2answers
43 views

Area of greatest integer function

Question: Find the area enclosed by the function: $$\left\lfloor\frac{\left|3x + 4y\right|}{5}\right\rfloor + \left\lfloor\frac{\left|4x - 3y\right|}{5}\right\rfloor = 3$$ where ...
1
vote
2answers
30 views

Is this an acceptable trig-sub and reversion from trig at the answer?

Today I had to take the indefinite integral $\int x^3 \sqrt{x^2-1} \, dx$ My steps: $x=\sec\theta$, $dx=\sec\theta\tan\theta\, d\theta$ $\displaystyle \int \sec^3\theta \sqrt{\sec^2\theta - 1} \, ...
1
vote
4answers
58 views

Evaluating integral with $e^{\sin x}$

I had this integral $ \int e^{\sin(x)} {\sin(2x)} dx$ I tried to split it up using integration by parts but I can't evaluate integral of $e^{\sin x}$
-2
votes
3answers
74 views

Is it possible to solve $\int_{0}^{1} t^4 \sqrt{1+t^2}\,dt$ [on hold]

How should I solve the below integral?$$\int_{0}^{1} t^4 \sqrt{1+t^2}\,dt$$
2
votes
1answer
27 views

How can I evaluate this contour integral?

Suppose we have the following contour integral, in the complex plane: $$ \int_{\gamma} \frac{e^{\frac{1}{z}}}{z^{2}} \; dz $$ where $\gamma (t) = e^{it}$ for $0 \leq t \leq 2 \pi$. To solve this, I ...
-1
votes
0answers
16 views

Integration of exponential and trigonometry [duplicate]

Please explain howto integrate this formula. Question Regards!
4
votes
2answers
43 views

Possible alternative for finding the Area under the floor function (aka, the integral of floor(x))

So, I had to ask myself the question as to what the area under the floor function could possibly be. I started by graphing the basic $\mbox{floor}(x)$ function (I personally use desmos.com for a nice ...
1
vote
1answer
51 views

An integral involving hyperbolic functions

$$ \large \displaystyle \int_0^\infty {\dfrac{e^{-2x} \tanh\frac{x}{2}}{x \cosh x}dx} = 2 \ln \frac{\pi}{2\sqrt{2}} $$ How to prove the above integral? What I tried : $\displaystyle I(s) = ...
0
votes
0answers
38 views

Integration by parts thrice [on hold]

To find real fourier coefficients this resolves to a triple integration by parts. When I let $x=u$ and $dv=e^xcosxdx$ to find the $cos(nx)$ coefficients, This translates to $u∫dv-∫∫dvdu$, My ...
2
votes
1answer
35 views

Integrating $ \frac{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| |\sin(x)| dx }{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| dx } $

I'm trying to integrate $ \frac{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| |\sin(x)| dx }{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| dx } $ I understand that $|\cos(x)|$ and $|\sin(x)|$ when integrated over $- ...
0
votes
0answers
15 views

Proof of the finiteness of integral (in option pricing)

I would like to ask for help with proving the finiteness of the following double integral. $$\int_{0}^{\infty}e^{\alpha+k}\int_{k+\zeta}^{\infty} (e^{-\zeta+x}-e^k)f(x)\ \mbox{d}x\ \mbox{d}k,$$ ...
0
votes
0answers
43 views

Show that, $\int_0^1\frac{1}{1+x^2}\sum_{n=1}^{\infty}x^{2^n-1}dx =\frac{2}{\pi}$ [on hold]

Show that, $$\int_0^1\frac{1}{1+x^2}\sum_{n=1}^{\infty}x^{2^n-1}dx =\frac{2}{\pi}$$
0
votes
1answer
25 views

Solving an integral by Cauchy Formula

I want to solve the integral $$\oint_{|z|=\frac{1}{2}}{\frac{e^{1-z}}{z^3(1-z)}dz}$$ Its a long time ago that I solved such integrals. Is it just by definition of the line integral? Maybe someone can ...
0
votes
2answers
73 views

Integral of $\frac{1}{( x^{2015} - x)}$

I am trying to find the integral of $\dfrac{1}{( x^{2015} - x)}$. Does anyone know how to do this? I can't possibly do a you substitutions right? Can't do partial fraction either.
2
votes
3answers
69 views

Determine the derivative $\frac{dy}{dx}$ of the integral

Determine the derivative of the integral $$ \,\int_{\sqrt x}^{0}\sin (t^2)dt $$ What does this question mean. I do not understand it and I think you can't integrate $\sin t^2\,$.
1
vote
0answers
21 views

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $ where $\sigma$ is the surface in the first octant made up of part of the plane $2x+3y+4z=12$ and triangular in the ...
0
votes
0answers
37 views

Find all possible values of the integral

Find all possible values of $\displaystyle I= \int_C \frac{dz}{1+z^2}$, where $C$ is a curve with initial point $0$ and final point $1$ that does not meet the poles of $\dfrac{1} {1+z^2}$. It looks ...
0
votes
1answer
16 views

Factorising Iterated Integrals

I've been doing a bunch of homework about iterated double and triple integrals recently and a bunch of solutions seem to skip steps and use shortcuts that everyone seems to know, but no one seems to ...
0
votes
0answers
33 views

Seeking help with finding the general solution of this differential equation

I am trying to find the general solution of the following equation. $$\int_0^\infty \frac{\partial f(x, t)} {\partial t} \sin(x \xi) \, dx = \xi \int_0^\infty f(x, t) \cos(x \xi) \, dx -\alpha \xi ^2 ...
0
votes
0answers
44 views

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal.

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal. The answer was $(0,1)$, but I don't get it. I tried to take the derivative of the function and equal it to $0$ ...
-3
votes
2answers
30 views

Triple Integral with spherical polar coordianates [on hold]

By changing into spherical coordinates (or by any other method) evaluate the triple integral $$\iiint_V xyz \ dxdydz,$$ where $V$ is the volume in $\mathbb{R}^3$ deifned by the inequalities ...
3
votes
0answers
62 views

A generalization of an integral related with $\zeta(2)$

It is pretty well-known (and not difficult to prove) that: $$ \int_{0}^{+\infty}\frac{x}{e^{x}-1}\,dx = \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} \tag{1}$$ but what is known about $$ I_2 = ...
1
vote
2answers
28 views

Specific example of integrating a 1-form over a curve

I was given the following definition in my course but no corresponding examples: Supppose $\gamma:[a,b]\rightarrow{M}$ is a smooth curve and $\omega$ a 1-form on $M$ (so $\omega:M\rightarrow{T^*M}$). ...
3
votes
3answers
87 views

Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $

I have to prove that given $\gamma=0.577216\ldots$, the Euler-Mascheroni constant, and $\pi=3.14159\ldots$, we have: $$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $$
1
vote
0answers
18 views

Double integral over an annulus

Question: Let $D$ be part of the annulus $1\le x^2+y^2 \le 4$ lying in the first quarter of the $oxy$ plane where $x \ge 0, y \ge 0$ and below the line $y=x$ Evaluate the integral $$\iint_D\ ...