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)

-1
votes
2answers
71 views

How to evaluate $\int_0^1 \frac{dt}{(t-\frac{1}{2})^2+ \frac{3}{4}}$ [on hold]

How to evaluate $$\int_0^1 \frac{dt}{(t-\frac{1}{2})^2+ \frac{3}{4}}$$
5
votes
0answers
20 views

Discrete analogue of Green's theorem

Following formula concerning finite differences is in a way a discrete analogue of the fundamental theorem of calculus: $$\sum_{n=a}^b \Delta f(n) = f(b+1) - f(a) $$ We can think about the Green's ...
1
vote
2answers
73 views

Solving $ \int_{0}^{2\pi} e^{-x} \lvert \sin x\rvert,dx $

Since it involves an absolute value, I assume I need to split it into two cases? For $ 0 \le x \le \pi $ $$ \int_{0}^{\pi} e^{-x} \sin x\,dx $$ and for $ \pi \le x \le 2\pi $ $$ \int_{\pi}^{2\pi} ...
1
vote
0answers
27 views

Existence of measure given the margin is a step function

Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function $$Q(x)=A, if \phantom{0}0\leq x < x^*$$ $$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$ s.t. $$A,B\in[0,1] ...
2
votes
0answers
31 views

Using residue theorem along a branch cut to evaluate the inverse Laplace transform

I am trying to find the inverse Laplace transform of $f(z)$ using the residue theorem. Can you please check to see if what am doing below is correct? I am not really sure about what I am doing. ...
5
votes
1answer
58 views

Using numerical methods to calculate integral

$$ \mbox{How can I go about calculating}\quad \int_{0}^{\infty}\,{\rm e}^{-100\,x^{2}}\,{\rm d}x\quad \mbox{to}\ {\sf\mbox{five}}\ \mbox{decimal places of accuracy ?.} $$ Do I use Simpson's Rule ?. ...
4
votes
4answers
56 views

Evaluate $\int\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm dx$

I have tried to evaluate $$∫\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm d x$$ using the following identity: $$\frac{d(\sin^{-1}{u})}{du} = \frac{du}{1+u^2}$$ So I then reformed the integral to this: ...
0
votes
0answers
34 views

Evaluate $h(z)=\frac {k}{2\pi} \int_CF(\theta)e^{ikz\cos \theta}\,d\theta$

Suppose this integral $$h(z)=\frac {k}{2\pi} \int_CF(\theta)e^{ikz\cos \theta}\,d\theta$$ $$0\le\theta\le\pi$$ $$|z|\le l$$ We are in complex $\theta$ plane. Assume we have knowledge of $F(\theta)$ ...
1
vote
3answers
71 views

Help evaluating $\int e^x \sqrt{1+e^{2x}}dx$ [duplicate]

$\int e^x \sqrt{1+e^{2x}}dx$ It's probably been answered somewhere, but I havent found it so far so I decided to post it as a question (if it has been answered point me in the right direction and I ...
2
votes
2answers
65 views

Evaluation of $\int\frac{1}{1+(x+1)^{{1}/{n}}}dx$ for $n\in \mathbb{N},$

Evaluate $$\int\frac{1}{1+(x+1)^{{1}/{n}}}\,\mathrm dx$$ for $n\in \mathbb{N}$ $\bf{My\; Try::}$ Let $$(x+1)=t^n\;,$$ Then $$dx = nt^{n-1}dt$$ So $$\displaystyle I = ...
0
votes
0answers
44 views

Laplace Transforms for Simple Model Rocket Thrust Equations

I spent some time this weekend establishing a thrust model for one of my model rockets as part of a school project that I have been working on. I came up with the following model: ...
3
votes
2answers
43 views

Differential Equation: $\frac{\mathrm{d} y}{\mathrm{d} x} = xy + y\sin x$

I'm trying to solve this differential equation and believe I may have solved it using the "separable equations" method. Here's my work: $$\frac{\mathrm{d} y}{\mathrm{d} x} = xy + y\sin x = y(x + ...
18
votes
2answers
988 views

Is indefinite integration non-linear?

Let us consider this small problem: $$ \int0\;dx = 0\cdot\int1\;dx = 0\cdot(x+c) = 0 \tag1 $$ $$ \frac{dc}{dx} = 0 \qquad\iff\qquad \int 0\;dx = c, \qquad\forall c\in\mathbb{R} \tag2 $$ These are two ...
4
votes
0answers
67 views

What is the indefinite integral of zero? [duplicate]

From the definition of indefinite integral I might say: Since the derivative of a constant is zero, thus the indefinite integral of zero is a constant. Therefore: $$ \frac{dc}{dx} = 0 \quad\iff\quad ...
0
votes
0answers
5 views

Explanation of the norm of the tagged partition going to zero

Could someone please explain me what it actually means for the norm of the tagged partition going of to zero. I know that this is part of the definition of the reimann sum but I also know that this is ...
1
vote
1answer
37 views

solving an ODE: problem with integration

I want to solve the ODE \begin{align*} - \left(|u'|^{p-2}u'\right)' & = 1 \quad \mathrm{in}\ (-a,a)\\ u(\pm a) & = 0 \end{align*} for $1<p<\infty$ and $a>0$. I thought I could do ...
-2
votes
1answer
34 views

Properties of $ f(k) =\int_0^k \frac{3+\sin(x)}{1+x^2}dx$

I have to prove that $ f(k) =\int_0^k \frac{3+\sin(x)}{1+x^2}dx$ is strictly monotonically increasing and bounded by $2 \pi$ . My idea was to show that $f'(x)$ is non-negative in the given interval ...
16
votes
5answers
246 views

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

Today I discussed the following integral in the chat room $$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$ where $0\leq a, b\leq \pi$ and ...
1
vote
1answer
33 views

Riemann integrability of a square of a continuous function

Let, $f(x)$ be continuous in $[0,1]$ such that, $\int_{0}^{1}x^{n}f(x)dx=0$ for $n=0,1,2,3,...$. Then prove that, $\int_{0}^{1}f^{2}(x)dx=0$. First we apply $1^{st}$ M.V.T. of integral calculus & ...
1
vote
1answer
53 views

how to solve complex integration problem

While working on complex integration problem I got stuck at the following problem: $\int \frac{|dz|}{|z-2|^2}$ where $|z| = 1$ is the domain. The only idea that I am getting is that we can use the ...
0
votes
0answers
46 views

Fourier series for logarithm of sine.

I looked up here: Fourier series of Log sine and Log cos I have modified the question: How can I derive the coefficient $a_n, b_n$ for $\log(\sin(x))$ in the fourier series representation? Also, I ...
-3
votes
4answers
105 views

Proving $\int x \, \mathrm{d}y =xy- \int y \, \mathrm{d}x$

How to justify formally that $$\int x \, \mathrm{d}y =xy- \int y \, \mathrm{d}x$$ Background: I don't know how to work with the $\mathrm{d}y, \mathrm{d}x$ and for me it's just a symbol for the ...
0
votes
1answer
16 views

Partial fraction decomposition coefficients problem [on hold]

How can I go about integrating $\int\frac{dx}{x(x-1)(x^2+4)^2}$ What are my factors?
4
votes
1answer
61 views

Reduction formula of primitive $\big(1-\sin^3{x}\big)^n\cos{x}$

I am trying to obtain a reduction formula for $$\int_0^{\pi/2}\big(1-\sin^3{x}\big)^n\cos{x}\;\mathrm dx $$ where $n \in \mathbb{N}$. My attempt is as follows $$\text{let } v = \sin{x}\; \implies ...
5
votes
0answers
64 views

Find a function that maximizes $\int_{0}^{1}f(x)\,\rm dx$ with given constraints

Find a function $f(x)$ that maximizes the following integral $$\max\int_{0}^{1}f(x)\,\rm dx\quad \text{s.t.}\quad \frac{d}{dx}ln(f(x))<0$$ $f(x)$ also continues, $f:[0,1]\rightarrow R$ and we ...
1
vote
1answer
20 views

Set up integral in spherical coordinates outside cylinder but inside sphere

I have the equation of a cylinder and the equation of a sphere given: Cylinder: $x^2+y^2=4$ Sphere: $x^2+y^2+z^2=25$ I'm asked to set this up in cylindrical and spherical coordinates. Cylindrical ...
1
vote
1answer
37 views

Growth of a sequence

Let $$a_n=\int_{\frac{\pi}{2}+n\pi}^{ \frac{3\pi}{2}+3n\pi}\frac{\cos{t}} {t} dt$$ How to show that $\left(a_{2n}\right)_{n\geq 0}$ is increasing (strictly) and $\left(a_{2n+1}\right)_{n\geq 0}$ is ...
4
votes
4answers
170 views

How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$

I found this nice result. Prove that $$\int_{0}^{\infty}\sin{x}\arctan\left({\frac{1}{x}}\right)\,\mathrm dx=\frac{\pi }{2} \left(\frac{e-1}e\right)$$ I tried some methods but I can't ...
0
votes
0answers
30 views

How to prove geometric mean is smaller than the arithmetic mean for a continuous distribution?

For discrete probability distribution, the geometric mean is defined as ${{\rm{E}}_{\rm{G}}}X = {\mu _G} = \sqrt[{\mathop \sum \limits_i {p_i}}]{{\mathop \prod \limits_i x_i^{{p_i}}}} = \mathop \prod ...
3
votes
1answer
46 views

Asymptotic elementary expression for the antiderivative of $x^x$

It is well known that there exists no elementary function $f$ with $$\int x^x\,dx \quad = \quad f$$ Is there an elementary function $g$ such that $$\int x^x\,dx \quad \tilde{} \quad g$$ in the ...
1
vote
1answer
18 views

Object moving on x axis integration

So, I think I know how to set up this problem, but then I get stuck at the last part: An object moves along the x - axis such that its velocity at time $t$ is $v(t) = cos(2t) $. Suppose the object ...
0
votes
0answers
17 views

Volume of a region R when revolved about the x-axis (multiple problems)

based on this earlier question I had Volume of a region R when revolved about the x-axis region bounded by y=x^2+1,y=2,x=0 is revolved about y=-1 region bounded by y=x/2, y=sqrt(2x) is revolved ...
4
votes
1answer
27 views

For what $p$ does the surface of revolution for $x^p$ have finite surface area?

I am trying to investigate the surface of revolution of the $x^p$ functions, in the domain $[1,\infty)$ Using the formula for surface of revolution, $$A=2\pi\int_1^\infty x^p ...
10
votes
4answers
123 views

Evaluating $\int_{0}^{\pi/2}\frac{x\sin x\cos x\;dx}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}$

How to evaluate the following integral $$\int_{0}^{\pi/2}\frac{x\sin x\cos x}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}dx$$ For integrating I took $\cos^{2}x$ outside and applied integration by parts. ...
2
votes
0answers
79 views

how to use Matlab ifft to calculate the following integral? [duplicate]

$$R(t)=\int_{-\infty}^\infty\dfrac{\omega e^{i\omega t}}{(3-\omega^2)^{2}+4\omega^2}\,d\omega$$ where t is a integer and $t>0$ I used to calculate this integral by numerical integral,but it seems ...
1
vote
1answer
29 views

Volume of a region R when revolved about the x-axis

Find the volume of the region when revolved about the $x$ axis $y= \sqrt{x-1}$, $y=2$, $y=0$, and $x=0$ Is this right? Also if you could help me with revolving this same region around $y=2$, ...
1
vote
2answers
44 views

If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ …

Suppose $f : [a,b] \to \mathbb{R}$ is continuous and $g \in \mathcal{R}[a,b]$ with $g(x) \ge 0$ for all $x \in [a,b]$. Show that there exists a $c \in [a,b]$ such that $$\int_a^b f(x)g(x) \, dx = ...
3
votes
2answers
41 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
39 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
70 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
125 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
89 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
31 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
187 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 ...