Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

Filter by
Sorted by
Tagged with
1
vote
2answers
311 views

Explain $-\int^{a}_{b}\frac{Q}{4 \pi \epsilon \bar{r}^{2}} \cdot d \bar{r}= \left[\frac{Q}{4\pi \epsilon \bar{r}}\right]^{a}_{b}$

Explain $-\int_{b}^{a}\frac{Q}{4 \pi \epsilon \bar{r}^{2}} \cdot d \bar{r}= \left[\frac{Q}{4\pi \epsilon \bar{r}}\right]_{b}^{a}$. What is the term $\frac{1}{\bar{r}^{2}} \cdot d\bar{r}$? I find it ...
5
votes
2answers
4k views

lebesgue integral uniform convergence

Let $f_n, f : [a,b] \to R.$ If $f_n \to f$ uniformly then show that the lebesgue integrals are equal. ie. $\int f = lim \int f_n$ This is clearly true for continuous functions, but how do I ...
3
votes
1answer
98 views

Finding the area

I was given the problem: find the area of the region bounded by $y=1/x$, $y=x^2$, $y=0$, and $x=e$. My approach was to break it up into two integrals, $\displaystyle \int_0^1 (x^2-0)\,dx$ and $\...
2
votes
2answers
372 views

arc length of $\dfrac{e^x - e^{-x}}{2}$

how can I please calculate an arc length of $\dfrac{e^x-e^{-x}}{2}$. I tried to substitute $\dfrac{e^x-e^{-x}}{2}=\sinh x$, which leads to $\int\sqrt{1+\cosh^2x}dx$, which unfortunately I can't solve. ...
1
vote
1answer
336 views

Integration by parts

Hi I want to integrate this integral and ask if my work is correct or not. $$\int^\infty_0 dx x^{\alpha-1} e^{-x} (a+bx)^{-\alpha}$$ I want to integrate it by parts, so I have $$(a+bx)^{-\alpha} = ...
2
votes
1answer
4k views

Definite Integral with unequal partitions

I am trying to understand how definite integral works when the partitions of the function are of unequal length. I found this link and am stuck here: $$I = \int_a^b f(x)dx = \lim_{mesh(P) \to 0} R(f,...
5
votes
3answers
388 views

How to compute the following definite integral

Studying some integral table, I came across the following definite integral $$\int_0^{\pi} \log [ a^2 + b^2 -2 a b \cos \phi ]\,d\phi$$ for $a,b \in \mathbb{R}$. Does somebody know a nice way to get ...
12
votes
5answers
4k views

Help solving $\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}dx}$

$\displaystyle\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}\,\mathrm{d}x}$ I used partial fractions, solved $A = 2, C = 3$. $$\frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} +\frac{(Dx+E)}{(x^...
7
votes
2answers
359 views

having trouble with limit of integral

How do I solve the following? $$\lim_{x\to 0} \int_0^1 \cos\left(\frac{1}{xt}\right)\, dt$$
2
votes
2answers
12k views

Integrate $x^ {1/2}e^{-x}$ using integration by parts

How to integrate $x^{1/2}e^{-x}$ using integration by parts? Answer should be $\left(-\sqrt{x} e^{-x}+(1/2)\sqrt{\pi} \mbox{erf}(\sqrt{x})\right)+c$
4
votes
2answers
548 views

Substitution Rule for Definite Integrals

I'm working on an integration by parts problem, and I'm trying to substitute to simplify the equation: $$\int_\sqrt{\frac{\pi}{2}}^\sqrt{\pi} \theta^3 \cos(\theta^2) d\theta$$ Using the substitution ...
1
vote
1answer
475 views

How to find the volume of the solid of revolution?

Q: Find the volume of the solid of revolution obtained by revolving the region bounded by the line $y = 2$ and the curve $y = \sec^2x$, $-90 \lt x \lt 90$, around the x-axis. Attempted solution: $(1/...
0
votes
1answer
137 views

Find the area bounded by these two functions?

Find the area bounded by these two functions: $$y = \frac{\ln x}{x}\quad\mbox{and}\quad y = \frac{1}{e} + \frac{(e^2+1)(x-e)}{e^2-1}.$$
0
votes
2answers
2k views

Find the area bounded by $y = \ln(x)$, $y = 1$, the x-axis and the y-axis

The answer in the book is $e-1$, and but I can't figure out how to go from $x(\ln(x) - 1)$ to that answer...
8
votes
5answers
7k views

Calculate $\pi$ precisely using integrals?

This is probably a very stupid question, but I just learned about integrals so I was wondering what happens if we calculate the integral of $\sqrt{1 - x^2}$ from $-1$ to $1$. We would get the surface ...
3
votes
2answers
929 views

What does it mean to say a function is differentiable with respect to lebesgue measure?

What does it mean to say a function is differentiable with respect to the lebesgue measure almost everywhere. A definition would be helpfull. Do I need to learn about the Radon Nikodym derivative to ...
2
votes
1answer
489 views

Equation with a definite integral - can I differentiate it?

I have an equation like this: $$te^{t} = \int\nolimits_0^t e^\tau u(\tau)d\tau$$ I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how ...
2
votes
2answers
160 views

A body falls through a medium

I'm hating these variable resistance questions. A body of mass $m$ falls from rest in a medium that produces a resistance of magnitude $m\cdot k \cdot v$. where $k$ is a constant, where the speed of ...
5
votes
3answers
577 views

Help solving another integral $\int (2x^2+4x-2)^{-\frac{3}{2}} \ dx$

$$\int (2x^2+4x-2)^{-\frac{3}{2}} \ dx$$ Complete the square? $$\int \frac{1}{(2(x+1)^2-4)^\frac{3}{2}} \ dx$$ Not sure what do do next, or if I should try something else? Big help if you can show ...
10
votes
3answers
966 views

How to find $\int_0^{\infty}\frac{dx}{(1+x^2)^4}$

How would you compute for the definite integral of $$\int_0^{\infty}\frac{dx}{(1+x^2)^4}$$ I know that integral of $\displaystyle \frac1{(1+x^2)}$ equals $\tan^{-1}x$. I tried using integration by ...
17
votes
4answers
1k views

Finding $\int_0^{\pi/2} \sin x\,dx$

I'm interested in why $$\int_0^{\pi/2} \sin x\,dx = 1.$$ I know how to do the integral the conventional way but am more interested in what makes radians special for this problem. If we instead compute ...
3
votes
1answer
189 views

Problem with integrating sine function

I have a problem solving an integration. This is my approach: \begin{align*} u &= \frac{1}{T}\cdot\intop_{0}^{T}(U_{0}+\hat{u}\cdot \sin(\omega t))dt\qquad\text{with }\omega=\frac{2\pi}{T}\\ u &...
1
vote
2answers
235 views

Show Integral Inequality

Suppose $m(E)=1$ and $f,g \geq 0$ with $fg \geq 1$. Show that $\displaystyle \int_E f dm \int_E g dm \geq 1$ Can someone please point me to the right direction?
7
votes
3answers
9k views

Evaluate $\int \cos^3 x\;\sin^2 xdx$

Is this correct? I thought it would be but when I entered it into wolfram alpha, I got a different answer. $$\int (\cos^3x)(\sin^2x)dx = \int(\cos x)(\cos^2x)(\sin^2x)dx = \int (\cos x)(1-\sin^...
2
votes
1answer
120 views

given p>1, whats an example of f where $\int_{-\infty}^{\infty} |f| < \infty$ but $\int_{-\infty}^{\infty} |f|^p = \infty$

given p>1, whats an example of f where $\int_{-\infty}^{\infty} |f| < \infty$ but $\int_{-\infty}^{\infty} |f|^p = \infty$ what about vica-versa? that is whats an example of g where $\int_{-\...
9
votes
3answers
4k views

Is there a closed form for $\int x^n e^{cx}\,\mathrm dx$?

Wikipedia gives this evaluation: $$ \int x^ne^{cx}\,\mathrm dx=\frac1cx^ne^{cx}-\frac nc\int x^{n-1}e^{cx}\,\mathrm dx=\left(\frac{\partial}{\partial c}\right)^n\frac{e^{cx}}{c}$$ But I have no idea ...
3
votes
3answers
847 views

Proving that there exists a unique f(x) given Area and arc-length of f(x) on a given interval

I've been suggested to this site by some nice people at mathoverflow.net Before I get started, let me tell you a little about myself. I’m a fourth year Mechanical Engineering student at the ...
11
votes
2answers
406 views

integration of a function

I found this explanation in a journal paper but I could not understand it. Can someone give me an explanation or possibly a proof that: If $$\frac{\mathrm{d}V(t)}{\mathrm{d}t}=\sqrt{2}\sum_{h=1}^{H}h\...
3
votes
2answers
656 views

Calculus Teacher being Tricky (Integrals)

So our calc teacher is being tricky and sending us off to fend for ourselves in the world of mathematics. Here's the question: We need to set up an integral to find the volume of the solid formed by ...
2
votes
2answers
409 views

Evaluating ${\frac{96}{6}}\int{\cos^4(16x)} \ dx$

$$\int{96\cos^4(16x)} \ dx$$ Setting $u=16x$, $du=16dx$, $${\frac{96}{16}}\int{\cos^4(u)} \ du$$ Kinda stuck here, I checked Wolfram Alpha but it suggests using some reduction formula that we haven't ...
1
vote
2answers
15k views

How to integrate the volume of a solid torus (donut-shaped solid)?

Please tell me if what I did is correct or if there's any faster alternatives. I set $x$ and $y$ axes on the center of the circle with radius $r$, therefore this can be seen as an area described by $...
5
votes
2answers
2k views

In what situations is the integral equal to infinity?

In the following integral, p(x) and q(x) are probability distributions. Can you help me to determine in what situation this integral is equal to infinity. For example, I think that such a situations ...
11
votes
2answers
2k views

Integration by parts: $\int e^{ax}\cos(bx)\,dx$

I need to evaluate the following function and then check my answer by taking the derivative: $$\int e^{ax}\cos(bx)\,dx$$ where $a$ is any real number and $b$ is any positive real number. I know that ...
3
votes
0answers
527 views

What is the expected value of modified Dirichlet distribution? (integration problem)

It is easy to produce a random variable with Dirichlet distribution using Gamma variables with the same scale parameter. If: $ X_i \sim \text{Gamma}(\alpha_i, \beta) $ Then: $ \left(\frac{X_1}{\...
14
votes
2answers
625 views

Proof of bound on $\int t\,f(t)\ dt$ given well-behaved $f$

I got the following question by mail from someone I don't know from Adam. (Quoted in part.) if $f(t)$ continuously diff. on $[0,1]$ and a) $\int_0^1f(t)\ dt=0$ b) $m\le f\,'\le M$ on $...
148
votes
19answers
23k views

Striking applications of integration by parts

What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your ...
3
votes
3answers
6k views

How to solve $\int\tan^3(x)\,dx$?

How to solve $\int\tan^3(x)\,dx$ ? Do I use substitution?
3
votes
7answers
1k views

How to deduce trigonometric formulae like $2 \cos(\theta)^{2}=\cos(2\theta) +1$?

Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^...
8
votes
3answers
3k views

Name of this identity? $\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$

Again: $$\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$$ Also the one for $\sin$: $$\int e^{\alpha x}\sin(\beta x) \...
5
votes
1answer
703 views

What's With The Integral $\int\sqrt{\cos(2\theta)}\, \mathrm d\theta$?

A student randomly asked me to compute $$\int\sqrt{\cos(2\theta)}\, \mathrm d\theta.$$ I was unable to do so, as were several other instructors. I typed the integral into Wolfram and it says that ...
3
votes
3answers
584 views

Integrating $\int\frac{2dx}{x\ln(6x)}$

I needed some help integrating this: $$\int\frac{2\,dx}{x\ln(6x)}.$$ I have never seen the dx within the problem like that, I am assuming I can't just move it to the outside can I? Can I start by ...
2
votes
2answers
1k views

Need help integrating $e^{8x}/(e^{16x}+36)$

I started off substituting $u = e^x$ so $du = e^x\, dx$, giving me $$\int \frac{u^8}{u^{16}+36}\,du.$$ So then I substituted $v = u^8$ and $dv = 8u^7du$, which gives me $$\int \frac{1}{v^2 + 36}\,...
3
votes
1answer
2k views

Does $\int_{\mathbb R} f(x)x^n dx = 0$ for $n=0,1,2,\ldots$ imply $f=0$ a.e.?

Let $f(x)$ be a real-valued function on $\mathbb{R}$ such that $x^nf(x), n=0,1,2,\ldots$ are Lebesgue integrable. Suppose $$\int_{-\infty}^\infty x^n f(x) dx=0$$ for all $n=0,1,2,\ldots. $ Does it ...
3
votes
4answers
1k views

How can I find $\int_{\sqrt2/2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}dydx$?

My question is ; How can I solve the following integral question? $\displaystyle \int_{\sqrt2/2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}dydx$ Thanks in advance,
7
votes
3answers
33k views

Integrating $e^{f(x)}$

can someone tell me a way of integrating functions like $e^{f(x)}$ I have a specific case: $\int e^{-3x}\,\mathrm{d}x$ PS: I'm not looking for the answer of this, but the way of doing it. Thanks ...
4
votes
1answer
108 views

Why isn't this simpler partition enough?

I'm trying to figure out a lecture example given on our Analysis course. We are currently going through Riemann integrals. Let $g:[0,1] \to R, g(x) = 1$ when $x \in [0, \frac{1}{2}]$ and $g(x) = 2$ ...
6
votes
3answers
13k views

Applying Green's Theorem

So I'm trying to solve this problem stated like this: Using Green's Theorem, find the area of the elipse defined by (where $a,b \gt 0$): $$\frac{x^2}{a^2} + \frac{y^2}{b^2} \leqq 1$$ I'm ...
2
votes
4answers
2k views

Approximating $\pi$ using Monte Carlo integration

I need to estimate $\pi$ using the following integration: $$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$ using monte carlo Any help would be greatly appreciated, please note that I'm a student trying to ...
3
votes
1answer
245 views

Do closed-form expressions exist for these integrals?

Playing with integrals on the form $$\int \frac{1}{1-x^n}\,dx$$ I noticed that for odd values of n > 5, it doesn't appear to be possible to express the integral as a closed-form expression. Is this ...
1
vote
0answers
555 views

Integration - changing the limits

I have an integral: $$\int_0 ^a\int_0 ^b\int_0 ^a\int_0 ^b \sin(x)\sin(\bar{x})\sin(y)\sin(\bar{y})f(x,\bar{x},y,\bar{y}) \, dx \, dy \, d\bar{x} \, d\bar{y}$$ Where: $$f= \dfrac{\sin(\sqrt{(x-\bar{x}...