All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

learn more… | top users | synonyms (3)

3
votes
2answers
42 views

How to find $\int \sec^{3} \ dx$ [duplicate]

I am stuck trying to find $$\int \sec^3{x} \ dx.$$ Here is my attempt using integration by parts: $$\int \sec^3{x} \ dx = \sec{x}\tan{x} - \int \tan^2{x}\sec{x} \ dx.$$ At this point, I am stuck. ...
1
vote
2answers
29 views

area under a curve and units

If we introduce a unit of length like meter for $x$ and integrate the function $f(x)=x^2$ from $0$ to $2m$ we get $\dfrac{8}{3} m^3$. How can this be interpreted geometrically? My initial thought was ...
1
vote
0answers
41 views

What does $dx$ mean in an indefinite integral?

$\int_0^{1} 3{x^3}\,{dx} $ $\int3{x^3}\,{dx} $ Both the definite and indefinite integrals have the same mark in their ends, ${dx}$. The latter one's ${dx}$ would mean an infinitely small width of ...
0
votes
1answer
40 views

Why do the limits of integration matter in a double integral?

Okay, I know that seems like a stupid question but I couldn't think of a better way to phrase it. I was trying to understand why iterated integrals involve "projecting" the domain onto one of the ...
4
votes
2answers
71 views

How to integrate $\ln \big( b + \sqrt{b^2 + c^2 + x^2}\,\big)$?

I am looking to demonstrate the following result. Any ideas are much appreciated. $$ \begin{align}\int \ln \left( b + \sqrt{b^2 + c^2 + x^2}\right) dx = &\;x \ln \left( b + \sqrt{b^2 +c^2 ...
4
votes
1answer
33 views

power series for $\int_0^x e^{-t^2}dt$

Use a known power series expansion to find the power series representation of the integral function $g(x) =\int_0^x e^{-t^2}dt$ centered at $a=0$ My approach Note that $g'(x) = e^{-x^2}$. ...
9
votes
3answers
131 views

How to find $\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$

Evaluate $$\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$$ where $a$ is a real parameter $a\geq1$. I can easily find the definite integral for $a=1$. It is $\sin(1)$. In wolframalpha.com when I put ...
3
votes
2answers
90 views

The integral of $e^{-x^2}$ [duplicate]

How can I integrate this by parts? It seems to become recursive. I'm familiar with the classical solution, and cannot use that here due to the constraints of this class. Here's the integral (to ...
2
votes
1answer
32 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
5
votes
1answer
190 views

Riemann Integrability in $\Bbb R^2$

Define the General Subdivision $S$ of a rectangle $R$ in $\Bbb R^2$ as a collection $E_1,...,E_k$ of Jordan regions such that none of them has interior points in common, and: $$R \subset ...
2
votes
0answers
12 views

Finding the Area of a Torus-like surface

I'm trying to find out the Area of the following surface: Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that ...
2
votes
1answer
18 views

A question regarding Surface Integrals and Stoke's Theorem

Let $G$ be an open set in $ \Bbb R^3$ and $F:G \rightarrow \Bbb R^3-{0}$ a vectorial field of class $C^1$. Suppose that $S$ is an open set, contained in $G$, whose non-empty boundary $\delta S$, is ...
0
votes
0answers
37 views

Solving a fractional trigonometric integral

On an answer on another SE I wrote, I came upon the integral $$\int \frac{x}{a-b\cos\left(\frac{x}{2}\right)+c\left(\cos^2\left(\frac{x}{2}\right)+\sin^2\left(\frac{x}{2}\right)\right)}dx$$ Solving it ...
2
votes
2answers
62 views

Supremum of a sine integral

Let $M_T=\int\limits_{0}^{T}\frac{\sin(t)}{t}dt$ be a sine integral. Why is $2\displaystyle\sup_{T}M_T < \infty$?
2
votes
1answer
60 views

I can't solve this integral in Mathematica

I want to calculate the following integral in Mathematica 10 $$f(x) = ...
1
vote
0answers
16 views

Decisions on the order of integration with double integrals (when Deriving PDF via CDF) (Bank Problem)

Consider the following problem: Gandalf, Saruman and Radagast go to a bank together. There are two open counters which Gandalf and Saruman immediately go to get their service. Radagast goes to the ...
0
votes
0answers
15 views

Weber Transform

During my studies I meet the Weber Transform of the free space potential function, that is: $$\int _{\rho }^{\infty }\exp(-i \text{$\lambda $p} t) (Y_0(\text{$\lambda $p} \rho ) J_0(\text{$\lambda ...
0
votes
0answers
20 views

Trig sub and Integration of Squareroot divided by polynomial squared

Question #2 What am I doing wrong? Do not give me the answer but rather a hint.
0
votes
1answer
34 views

Evaluate integral by interpreting it in terms of areas

I tried (a) and I got 5, but I am suppose to get a 4. I really need a good explanation to understand how to approach these problems. I tried searching in youtube and stuff, but it was not helpful. ...
5
votes
1answer
66 views

An advanced integral $\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$

I'd like to ask you how you would like to approach the integral below $$\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$$ and then recommend me some tools you'd employ. It's ...
0
votes
0answers
31 views

Rewrite the integral $\int_{0}^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)\,dz\,dy\,dx$ in the orders $dx\,dy\,dz$ and $dy\,dz\,dx$.

Rewrite the integral $$\int_{0}^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)\,dz\,dy\,dx$$ in the orders $dx\,dy\,dz$ and $dy\,dz\,dx$. My try: We have $z=0,z=1-y,y=\sqrt{x},y=1,0\leq x\leq 1$ ...
0
votes
0answers
42 views

Even and odd integrals

Find the definite integral $$\int_{-2}^{2} \Big(2f(x) + 3g(x)\Big)dx$$ where $f(x)$ is an even function such that $$\int_{0}^{2} f(x)dx = 3$$ and $g(x)$ is such that $$\int_{-2}^{4} g(x)dx = -3 ...
1
vote
1answer
34 views

Area within $x=0$ $y=x$ and $e^{-x}$

Is there a way to find the area within $x=0$ $y=x$ and $e^{-x}$ without solving numerically $e^{-x}=x$ ?
0
votes
1answer
27 views

Integral of Euler's formula

Why is $=\int\limits_{-\infty}^{\infty}\cos(-tx)dF(x)+i\int\limits_{-\infty}^{\infty}\sin(-tx)dF(x)=\int\limits_{-\infty}^{\infty}\cos(tx)dF(x)-i\int\limits_{-\infty}^{\infty}\sin(tx)dF(x)$? I know ...
5
votes
3answers
181 views

How to evaluate $\int_{0}^{\infty}\frac{(x^2-1)\ln{x}}{1+x^4}dx$?

How to evaluate the following integral $$I=\int_{0}^{\infty}\dfrac{(x^2-1)\ln{x}}{1+x^4}dx=\dfrac{\pi^2}{4\sqrt{2}}$$ without using residue or complex analysis methods?
0
votes
1answer
14 views

Iterated integral over non-rectangular region problem.

Could we integrate $S$, where $S$ is the region bounded between $y = 1$ and $y = x^2$ over the function $xy$ by taking $y$ constant? I solve the problem by taking $x$ constant and get the result $0$. ...
1
vote
0answers
45 views

Proving $\displaystyle{\int \frac{x^3}{{\rm e}^x - 1}}dx$ cannot be evaluated in closed form in terms of elementary functions

It is known the following indefinite integral $$\int \frac{x^3}{{\rm e}^x - 1} dx$$ cannot be evaluated in closed form in terms of any of the elementary functions of mathematics. A proof of this can ...
1
vote
3answers
60 views

Integration of $1/(x\sqrt{25x^2-1})$

$$\int{\frac{1}{x\sqrt{25x^2-1}}}\,dx$$ Let $x=\frac{1}{5}u$ Now when I substite it and simplify I get $$\int{\frac{1}{u\sqrt{u^2-1}}}\, du$$ There is a trig identity which says that this is equal ...
6
votes
3answers
433 views

A Sum that came up while solving a integral

While evaluating $I$, I did the following- $$\begin{align}I= \int_{0}^{1} \log \left(\dfrac{1+x}{1-x}\right) \dfrac{1}{x\sqrt{1-x^2}} \ \mathrm{d}x &= 2 \int_{0}^{1}\sum_{n=0}^{\infty} ...
-1
votes
1answer
34 views
2
votes
2answers
61 views

Limit of an integral

I'm not sure how to approach (no pun intended) the following limit: $$\lim_{x \to 0^{+}} \sqrt{|\sin x - \tan x | } \int_{\cos x}^{1+ \sin x} e^y \, \, \mathrm{d}y$$ I know that the indefinite ...
3
votes
5answers
97 views

Trigonometric substitution and Integration of $\frac{1}{x^2\sqrt{x^2+1}} $

Regarding the integral $$ \int \frac{dx}{x^2\sqrt{x^2 + 1}} $$ I'm not sure what to do about the extra $x^2$ in the denominator. What can I do about it?
0
votes
2answers
36 views

Integrating exponential function with elliptic bounds

I am trying to integrate the following: $$\iint_R\exp\left(\frac{x^2}{4}+\frac{y^2}{16}\right)\:\mathrm{d}A$$ With the region $R$ having the bounds: $$\frac{x^2}{4}+\frac{y^2}{16}=3$$ ...
2
votes
2answers
72 views

A reduction formula for $\int_0^1 x^n/\sqrt{9 - x^2}\,\mathrm dx$

Let $$I_n = \int_0^1 \frac{x^n}{\sqrt{9 - x^2}}\,\mathrm dx$$ Using integration, show that $$nI_n = 9(n - 1)nI_{n - 2} - 2\sqrt2$$ I've found that $\displaystyle I_0 = ...
-1
votes
3answers
39 views

Solve $\int \frac{30}{(x^2+9)^2}dx$ [closed]

Solving the Integrate $$\int \frac{30}{(x^2+9)^2}dx$$ help me with this excercises.. i need 1 idea, please
0
votes
1answer
23 views

Riemann integrable proof and notation

For Riemann integrable proof, I see $f \in \Re(\alpha)$. Also I see $U(p,f,\alpha)$. What does $\alpha$ stand for? Also to prove Riemann integrability, what do I do at very first step? I know my ...
1
vote
5answers
74 views

Solving $\int \frac{6x+3}{x^2+9}dx$ [closed]

Solving the Integrate help me with this excercises.. $$\int \frac{6x+3}{x^2+9}dx$$
0
votes
3answers
21 views

stuck on integrating fractions for ODE

I'm working on the ODE $\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}$ I factored it $\frac{dy}{dx}=\frac{(x-1)(y+3)}{(x+4)(y-2)}$ I used separation of variables $\int \frac{y-2}{y+3} dy = \int ...
0
votes
2answers
39 views

Prove that $\frac{1}{2}ab \equiv \int_0^b \! f(x) \, \mathrm{d}x$ when calculating the area of a right triangle.

Triangle $ABC$ is a right triangle with sides $AB$, $BC$ and $AC$. $a$ is the length of $AB$. $b$ is the length of $BC$. $c$ is the length of $AC$. If $a = 3$, and $b = 4$, we can use ...
0
votes
2answers
56 views

Solve: $\int \frac{x+4}{x^2-4x+3}$ [closed]

Solving the Integrate $$\int \frac{x+4}{x^2-4x+3}$$ can you help me with this excercises..
0
votes
1answer
39 views

How to determine $\int\cot^4(x) dx$ [closed]

How do I determine $$ \int\cot^4(x) dx $$
0
votes
3answers
68 views

Find $\int\sin^4(x)\cos^2(x)\,dx$

Find $$\int\sin^4(x)\cos^2(x)\,dx$$ My Attempt: $$\int\sin^4(x)\cos^2(x)\,dx = \frac18 \int ((1-\cos(2x)-\cos^2(2x)+\cos^3(2x))\, dx$$ How to proceed from here?
-7
votes
0answers
80 views

Need help to find $\int_0^\infty \frac{e^{-x^2}\sin x^2}{\ln(1+x^2)}dx$ [closed]

What is $$\int_0^\infty \frac{e^{-x^2}\sin x^2}{\ln(1+x^2)}dx?$$ It is quite tempting to suggest a substitution as $y=\ln(1+x^2)$, but it doesn't seem that such would work out well... Numeric value ...
0
votes
0answers
13 views

Area of a smooth parametrized surface

Let $S=\mathbf{X}(D)=(x(s,t),y(s,t),z(s,t))$ be a smooth parametrized surface with $(s,t) \in\mathbf{R}^2$, where $D$ is the union of finitely many elementary regions in $\mathbf{R}^2$ and ...
0
votes
1answer
25 views

Prove the series of functions converges uniformly at $[-a,a]$ where $0<a<1$.

Let $$ \sum_{n=0}^\infty \left( \frac{x^{2n+1}}{2n+1} - \frac{x^{n+1}}{2n+2} \right) $$ Prove the series converges uniformly to $\frac{1}{2}\log(x+1)$ at $[-a, a]$ where $0<a<1$. I've ...
-1
votes
1answer
86 views

Evaluate $\int\left(1+x^2\right)e^{\arctan x}\,dx$ [closed]

How does one evaluate the following integral $$\int\left(1+x^2\right)e^{\arctan x}\,dx$$ Thanks.
-1
votes
1answer
18 views

Limit of integral of Lebesgue integrable function [closed]

Let $f:\Bbb{R}\to\Bbb{R}$ be a Lebesgue integrable function. Is $ \lim\limits_{n\to\infty}\int_{\lvert x\rvert\geqslant n} f(x) dx = 0 ?$
2
votes
0answers
26 views

Determining solid region from bounds of triple integral

If you have an integral such as: $$\int_0^1\int_0^{2-x^2}\int_0^{2-x}f(x,y,z)dydzdx$$ How can you determine the equation for the solid region represented by the bounds of this triple integral? Does ...
3
votes
2answers
73 views

How to evaluate $\int_0^1 \frac{2-t}{t^2-t+1} dt$?

How to evaluate $$\int_0^1 \frac{2-t}{t^2-t+1} dt\;?$$ I tried doing it using $s=-t+1$ but it wasn't useful. We've learned in class that having a polynomial in the denominator is considered to be ...
0
votes
0answers
50 views

Definite integral involving Error function

Let us write $$\mathrm{erf}(x)=\frac{2}{\sqrt {\pi}}\int_0^x e^{-t^2}dt $$ for the usual Gauss error function. Given natural numbers $m,n,k$ I am interested in computing the integral ...