0
votes
0answers
22 views

Evaluating $\int_1^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(x)\:\mathrm{d}y\:\mathrm{d}x$ using polar coordinates?

How is the following integral found using polar coordinates. $$\int_1^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(x)\:\mathrm{d}y\:\mathrm{d}x$$ I know the the part of the domain the circle being asked in ...
0
votes
1answer
29 views

Don´t know how to start proving this formula.

\begin{equation*} \int \frac{\cos ^{m}x}{\sin ^{n}x}dx=-\frac{\cos ^{m+1}x}{(n-1)\sin ^{n-1}x}- \frac{m-n+2}{n-1}\int \frac{\cos ^{m}x}{\sin ^{n-2}x}dx+C,\qquad (n\neq 1). \end{equation*} I`d like to ...
1
vote
4answers
43 views

Are discontinuous functions integrable? And integral of every continuous function continuous?

According to me answer of second part is yes as integration simply means area under curve.
2
votes
2answers
30 views

Integration by reduction

I have learnt how to integrate by reduction formula but this one seems to give me hell someone to lift me by telling me what to do or simply to solve it. \begin{equation} I_n=\int\sec^n x\,dx ...
2
votes
2answers
104 views

Why do we bother with $u$-substitution?

This question has bothered me ever since I learned $u$-substitution (A note here: I have no formal education at this level, so I may definitely have missed something). The method is presented as an ...
0
votes
3answers
40 views

How to solve integration of $\int x(x^2+k^2)^{-1/2} \, dx$?

As said in title, how do you solve integral $\int x(x^2+k^2)^{-1/2}\,dx$ where $k$ is some constant?
1
vote
1answer
35 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
1
vote
1answer
43 views

Suppose $f(x)\in L_1$ - Prove that $\lim_{n\rightarrow\infty}\int_0^\infty f(x)\cos(nx)dx = 0$

Assuming knowledge of the cyclic behavior of $cos(x)$, integration by parts, and $\int_0^{\infty} f<\infty$ is enough here? Consider \begin{align} & \int_0^\infty f(x)\cos(nx)dx = ...
0
votes
1answer
43 views

Are there integrals you can't solve without inverse hyperbolic substitution?

Are there any integrals that can't be solved with only trig substitution? An integral that requires you to use a hyperbolic or inverse hyperbolic substitution?
0
votes
3answers
42 views

How to find $\int\sqrt{(26x-x^2)}dx $

How do I find $\int \sqrt{(26x-x^2)} dx $ Is this an integration by parts question? Thanks, --Nick
1
vote
2answers
83 views

Equality of integrals: $ \int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \int_{0}^{1} \frac {1}{1+x^2} \, \mathrm{d}x $

In Street-Fighting Mathematics (page 16), Prof. Sanjoy Mahajan states that $$ \displaystyle\int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \displaystyle\int_{0}^{1} \frac {1}{1+x^2} \, ...
2
votes
2answers
55 views

Integral $I=\int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx$

Hi I am trying to show $$ \int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx=\frac{\pi}{32\alpha^{3/2}},\quad \Re(\sqrt \alpha)> 0. $$ I am looking for a solution to this NOT using contour integration, but ...
1
vote
0answers
17 views

difference of the values of a function is an integral

This is a very simple quesiton but something I don't understand. From Taylor expansion: $$f(y)-f(x)=f'(x)(y-x)+O((y-x)^2)$$ so, if I just picture that, on the left is the difference between two values ...
3
votes
2answers
75 views

Evaluate $\int x \sqrt{1 - x^4} \,\mathrm{d}x$

I have the following question $$\int x \sqrt{1 - x^4} \,\mathrm{d}x$$ I know we have to use trig. substitution for this and therefore, I did the following by letting $x = \sin \theta$ and $dx = \cos ...
1
vote
0answers
13 views

Prove with Lebesgue’s Criterion for integrablility that the composition $f\circ g$ is integrable

I have this homework question regarding Lebesgue's criterion for integrability and could use a bit of help. I'm not sure if my proof is entirely correct or formal enough. Here is said question: ...
6
votes
4answers
665 views

Formula for computing integrals

For computing derivative of a function, we can use the definition of a derivative, i.e. $$\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}.$$ Is there some for computing integrals too?
1
vote
0answers
42 views

Cauchy Integral Theorem problem (lack of understanding)

First of all i was asked to evaluate this integral $\int_\gamma \frac{2z}{(z-1)(z-3)} dz$ where $\gamma (t) = 2e^{it}$ for $0\leq t \leq 2\pi$. Now I thought I would have to calculate this ...
2
votes
1answer
20 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
-2
votes
1answer
40 views

Integrate $\cot^2x-\frac{\cos^2x}{\tan^2x}$ [on hold]

Integrate $\int{\cot^2x-\frac{\cos^2x}{\tan^2x}}dx$
-2
votes
1answer
45 views

Integrate $\int^{1}_{0}{\sin^2x}$ [on hold]

What is the value of this integration ? $\int^{1}_{0}{\sin^2x}dx $
2
votes
1answer
48 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
5
votes
2answers
96 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
0
votes
0answers
6 views

Find a map T(D*)=D and Triple integral

D={(x,y,z)| (7x-3y-z)^2 +(-3x+7y-z)^2 +(-x-y+3z)^2<=100} D* = {(u,v,w)|u^2+v^2+w^2<=1} find map T(D*)=D express the triple integral of xy dx dy dz over D as an integral over D* and evaluate
1
vote
0answers
30 views

Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$

Hi I'm trying to show that $$ I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}. $$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
7
votes
1answer
107 views

Calculate the following Integral (Please Help)

I am trying to calculate: $$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$ I am not looking for an answer but simply a nudge in the right direction. A stradegy, just something that would get me started. ...
0
votes
1answer
26 views

How do I integrate this in terms of error function

How do I evaluate $$\dfrac{1}{\sqrt{4\pi t}}\int_0^{\infty}ye^{-\frac{(\xi-y)^2}{4t}}dy$$ in terms of $\text{erf}(x)$ ? I tried integration by parts but the integral seems to get complicated. I think ...
5
votes
1answer
62 views

How to evaluate $\int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt$?

$$ \int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt $$ I tried doing it by parts and looking for differentials but I just keep getting back to the original expression. I can't think of a clever ...
1
vote
1answer
46 views

integral $I=\int_{-\infty}^\infty e^{-\alpha x^{2k}}dx$

$$ I=\int_{-\infty}^\infty e^{-\alpha x^{2k}} dx $$ The last problem was ill posed, and is answered in the post! You can disregard this post!
1
vote
1answer
29 views

Integrating an equation with both cos and tan

$$\int2\cos^5x\cdot\tan^6x\cdot dx$$ $$2\int\cos^5x\cdot\frac{\sin^6x}{\cos^6x}\cdot dx$$ $$2\int \frac{\sin^6x}{\cos{x}} dx$$ $$2\int\cos^{-2}x\cdot \sin^6x\cdot \cos{x}\cdot dx$$ ...
0
votes
1answer
66 views

Integral of $\sin|x|$

$$\int\sin|x|~dx$$ We have two cases: x less than zero, or x equals or higher than zero. $$\int_{-\infty}^0\sin(-x)~dx+\int_0^\infty\sin x~dx$$ Left side of this sum is equals to right side, so we ...
0
votes
3answers
30 views

How to get from $3\int_{-1}^0 (x^3-x) dx \,\,\,- \,\,\, 3\int_0^1 (x^3-x) dx$ to $6\int_{-1}^0(x^3-x)dx$?

Homework problem: Set up the definite integral that gives the area of the region. Two functions are given: $y_1 = 3(x^3-x)$ $y2 = 0$ The graph of $y1$ runs from x=-1 to x=1. I've gotten this ...
0
votes
2answers
46 views

Methods to do Integral

I know this integral can be done using complex analysis. Are there some slick solutions using standard calc methods? $\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{1}{(x^2+1)(x^2+9)}dx$
2
votes
0answers
41 views

Is there a generalization of integration by parts?

here is what i concerned: there are $u(x)$ and $v(x)$ in the original integration by part formula, what if the integral involve with one more function $w(x)$. Second of all, i know that there are ...
1
vote
1answer
33 views

Bessel's integral, how to actually evaluate?

I am just about to study Bessel functions and I have recently seen one of its integral representations given by: $$ J_ \alpha (x) = \frac{1}{\pi} \int_0 ^ \pi \cos(\alpha \tau - x\sin\tau) d\tau - ...
0
votes
1answer
23 views

Changing the domain of integral

I am studying how we use polar substitution to solve double integrals. However, I am struggling with finding the correct limits of the transformed integrals to obtain a suitable solution. eg: Why ...
1
vote
2answers
35 views

Integrating a Partial Derivative

Would I be right to think that $$\int dx \,\,\,\frac{\partial}{\partial x} f(x,y)=f(x,y)$$ Or are there pathological cases?
3
votes
1answer
62 views

Does this integral have any closed form? $\displaystyle\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$

Does this integral have any closed form? $$\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$$ I think the substitution $x=(u-1)+2\pi$ will do it, no?
2
votes
2answers
53 views

Evaluating integral limit in two ways gives different limits

Problem Show that $$\lim\limits_{h \rightarrow 0^{+}} \int_{-1}^{1} \frac{h}{h^{2}+x^{2}} \, dx = \pi.$$ I can do this by evaluating the integral directly and showing that it is equal to ...
1
vote
0answers
15 views

Question concerning the domain of polar coordinate.

So in the problems I encountered, I find it confusing about the domain of $\theta$. Problems take the form: For arbitrary function $f(x,y)$, and $$\displaystyle \iint_S f(x,y)dxdy=\iint_T ...
0
votes
0answers
19 views

Mean value of a function over the n-sphere superficie.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
0
votes
2answers
27 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
4
votes
2answers
63 views

Help finding a method for integrating this trigonometry function

I was going though a given proof in my book and part of it requires solving $$\frac{ad-bc}{2 \pi} \int^{2\pi}_0 \frac{d \theta}{(a \cos\theta+b\sin\theta)^2+(c \cos\theta+d\sin\theta)^2} $$ Which it ...
2
votes
1answer
33 views

A bounded integral

I want to show that there exists $K\in\mathbb{R}^+$ such that $$\left|\int_{1}^x \sin(t+t^7)dt \right|<K$$ for all $x\ge 1$. Intuitively, I'm quite sure it is true, but I can't find a formal proof. ...
1
vote
1answer
17 views

How do I calculate the area under a curve using the midpoints of rectangles?

I figured out how to calculate the area under the curve from the Right endpoint and Left endpoints, but I can't figure out how to calculate it using the midpoints. Especially when it says $M_3$. Ill ...
1
vote
1answer
22 views

Integral Application Word Problem

I'm given a problem that says a rock hammer was thrown at an upward initial velocity of 4.8m/s and had an initial height of 1.8m. The acceleration (on the moon) is 1.8m/s^2. What is the maximum ...
1
vote
1answer
107 views

Difficult Improper Integral

Evaluate the improper integral $$\int_0^\infty\frac{-38x}{(2x^2+9)(3x^2+4)} dx $$ I thought about doing this through partial fractions decomposition. However, when I tried, I got some really ...
0
votes
0answers
37 views

Line Integral and Green's Theorem

I have been working on a simple line integral: line integral of $x\,dy+y\,dx$ (I don't know how do write this properly, I'm sorry!) over the closed curve enclosed by the the ellipse $x^2+5y^2=4$ and ...
10
votes
1answer
92 views

Why is an equation necessarily dimensionally correct?

I have just read a fascinating proof of the value of the integral $$ \int_{-\infty}^\infty e^{-ax^2} dx, $$ which proceeds by dimensional analysis, as follows: we know that we can write $$ ...
0
votes
0answers
42 views

LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)d\theta$

I am trying to integrate the Log Sine Integral: $$ Ls_{n+1}=-\int_0^{\pi/3}\bigg[\ln\big(2\sin\frac{\theta}{2}\big)\bigg]^nd\theta $$ where n is a non-negative integer. This problem is strongly ...
1
vote
0answers
31 views

LogSine Generating Fn $ \int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...