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 ...

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4
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1answer
215 views

the integral :$ \int^\infty_{-\infty}e^{-x^{2}+x}dx $

$$ \int^\infty_{-\infty}e^{-x^{2}+x}dx $$ May be completing the square on the $-x^{2}+x$. Then, make a sub. then end up with the Gaussian integral with a constant multiple of $e^{\frac{1}{4}}$ ??
4
votes
1answer
115 views

A trivial problem in calculus

Try to compute $$\int\frac{dx}{x\ln x}$$ I compute it this way: first we have $x>0$. \begin{align*} \int\frac{dx}{x\ln x} &=\int\frac{d(\ln x)}{\ln x}\\ &=\ln|\ln x|+C \end{align*} But the ...
4
votes
3answers
21k views

How do I know which method of revolution to use for finding volume in Calculus?

Is there any easy way to decide which method I should use to find the volume of a revolution in Calculus? I'm currently in the middle of my second attempt at Calculus II, and I am getting tripped up ...
4
votes
2answers
238 views

To find closed form of $\int_0^{\frac{\pi}{2}} e^{-x\tan t+\alpha t} \;dt $

Let $x\geq 0$, then $$\int_0^{\frac{\pi}{2}} e^{-x\tan t+\alpha t} \;dt = U_{\alpha} (x) $$ $$-\int_0^{\frac{\pi}{2}} \tan t \ e^{-x\tan t+\alpha t} \;dt = \frac{d (U_{\alpha} (x) )}{dx} $$ ...
4
votes
1answer
648 views

Lebesgue integral (existence and finiteness) of $\sin(1/x^2)$

I think I am getting a little better at these MCT, DCT-type exercises. The issue is to show/prove the existence and finiteness (if they apply) to the following function: $$f(x)=\sin ...
4
votes
1answer
583 views

Evaluating Integral with Residue Theorem

The integral in question is $$\int_{_C} \frac{z}{z^2+1}\,dz,$$ where $C$ is the path $|z-1| = 3.$ The two pole of $f(x)$ where $f(x)=\frac{z}{z^2+1}$ is $-j$ and $j$ $${\rm ...
4
votes
3answers
104 views

Finding the integral of $\int x\ln(1+x)dx$

I know I have to make a u substitution and then do integration by parts. $$\int x\ln(1+x)dx$$ $ u = 1 + x$ $du = dx$ $$\int (u-1)(\ln u)du$$ $$\int u \ln u du - \int \ln u du$$ I will solve the ...
4
votes
2answers
97 views

Show $\int_\frac{1}{3}^\frac{1}{2}\frac{\operatorname{artanh}(t)}{t}dt=\int_{\ln 2}^{\ln 3}\frac{u}{2\sinh u}du$

How would I show (or explain) that $$\int_\frac{1}{3}^\frac{1}{2}\frac{\operatorname{artanh} t}{t}dt,$$ $$\int_{\ln 2}^{\ln 3}\frac{u}{2\sinh u}du,$$ and $$-\int_\frac{1}{3}^\frac{1}{2}\frac{\ln ...
4
votes
2answers
31k views

Integral of an absolute value function

How do I find the definite integral of an absolute value function? For instance: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$
4
votes
1answer
637 views

How does one integrate Landau symbols?

I have some big O()'s in an integral. How can i compute or estimate such an integral?
4
votes
2answers
320 views

Showing $\int_0^{2\pi} \log|1-ae^{i\theta}|d\theta=0$

This is a homework problem for a second course in complex analysis. I've done a good bit of head-bashing and I'm still not sure how to solve it-- so I might just be missing something here. The task is ...
4
votes
3answers
182 views

Finding an indefinite integral

I have worked through and answered correctly the following question: $$\int x^2\left(8-x^3\right)^5dx=-\frac{1}{3}\int\left(8-x^3\right)^5\left(-3x^2\right)dx$$ ...
4
votes
1answer
114 views

Integral two solutions

at school, we solved this integral and the solution we got was Wolfram Alpha gave a different solution Are these two solutions equal?
4
votes
2answers
568 views

Integral( using Euler substitution)

I try to solve this integral, but without success. Can you help me please? $$\int \frac{1}{2x+\sqrt{4x^{2}-x+1}}dx$$ Thanks a lot!
4
votes
1answer
247 views

How do you integrate $\cos(x^n)$, specifically for $n=-1$?

How does one integrate $\cos(x^{-1})$? I understand that the function is not defined at zero, but it is well defined, continuous, and real over the rest of $\mathbb{R}$. Nonetheless, when I put ...
4
votes
1answer
344 views

Function such that its square is not integrable

I have some set $A$ of Lebesgue measure $\mu(A)=1$. Does this imply that there is some measurable function $f: \mathbb{R}^n \to \mathbb{R}$ such that $$\int_A |f| d\mu< \infty, \int_A |f|^2 d\mu= ...
4
votes
1answer
178 views

Need help with line integral

Over a curve $C$ given by $(x^2+y^2)^2=30^2(x^2-y^2)$, What is $$ \oint\limits_C |y|\,\mathrm ds. $$ I've tried working on it but I couldn't get the solution. Here's how I did it: Using polar ...
4
votes
1answer
113 views

Can the difference between these integrals and certain related sums be expressed in a simpler way?

Suppose I have either of the following expressions: $$\newcommand{\rd}{\mathrm{d}} \int_1^n\int_1^\frac{n}{x} \,\rd y\, \rd x - \sum_{x=2}^n\sum_{y=2}^\frac{n}{x}1 $$ $$ ...
4
votes
1answer
603 views

The Cauchy-Crofton Formula

I am trying to understand the most basic formula from integral geometry. I have been looking at this website. The problem is things aren't working out for very simple examples. The circle works ...
4
votes
1answer
184 views

Please help with differentiation under the integral

This question has an answer that relates differentiation under the integral to the OP. Again, here's the original integral: $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$ ...and we let $$ F(y) = ...
4
votes
3answers
1k views

Young inequality

I am trying to prove young's inequality for integrals $$ ab \leq \int_0^a \! f(x) \, \mathrm{d}x + \int_0^b \! f^{-1}(x) \, \mathrm{d}x. $$ Can you help me please?
4
votes
3answers
100 views

Stuck on this integration $\int_0 ^\infty \frac{1}{1+x^2} cos(kx) dx =\frac{\pi}{2}e^{-k}$ [duplicate]

I'm not sure how to show this $$\int_0 ^\infty \frac{1}{1+x^2} \cos(kx) \ \mathrm dx =\frac{\pi}{2}e^{-k}$$ I tried by parts but I'm not getting anywhere, I'd really appreciate the help
4
votes
1answer
26 views

Does picking $C=0$ as a constant of integration result in a nominated anti-derivative?

Introductory calculus students are often introduced to the "indefinite integral" or anti-derivative before actually doing integrals because it makes the FTC seem natural (by some rather sketchy ...
4
votes
5answers
60 views

Find $f$ such that $\int_{-\pi}^{\pi}|f(x)-\sin(2x)|^2 \, dx$ is minimal

Fairly simple question that's been bothering me for a while. Supposedly it should be simple to solve from the properties of inner product but I can't seem to solve it. Find $f(x) \in ...
4
votes
3answers
183 views

Evaluation of $\int\frac{1}{x^4-5x^2+16}dx$

Evaluation of $\displaystyle \int\frac{1}{x^4-5x^2+16}\,dx$ $\bf{My\; Try::}$ Given $$\displaystyle \int\frac{1}{x^4-5x^2+16}dx = ...
4
votes
3answers
131 views

Integral $\int_0^\infty\sin{(x^4)} dx$

Can anyone help with how to evaluate the following integral $$ \int_0^\infty\sin{(x^4)} dx $$ I know that I need to use the fact that $\int_0^\infty e^{-x^4}dx=\Gamma\left(\dfrac{5}{4}\right)$ and I ...
4
votes
4answers
67 views

Integral $\int_{-\infty}^{\infty} \frac{1}{\sqrt{x^2+a^2}}\,dx$

$$\int_{-\infty}^{\infty} \frac{1}{\sqrt{x^2+a^2}}\,dx$$ I subbed in $x=a \tan\theta$ and ended up with $\ln|\sec\theta+\tan\theta|$. Is this correct? Thanks.
4
votes
1answer
96 views

Complex integration $\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$

I'm trying to evaluate the integral $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$$ using complex numbers. Meaning, instead of calculating $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt,$$ I want ...
4
votes
3answers
93 views

Convergence of $\int_0^\infty $sin$ (x^p) dx$

Consider the $\displaystyle \int_0^\infty $sin$ (x^p) dx$. Does it converge when $p<0$? Does it converge when $p>1$? My Work: Let $x^p=y$, then $\displaystyle \int_0^\infty $sin$ ...
4
votes
2answers
106 views

Integral of an exponential

I have the following: $$ I(a,b) \equiv\int_{-\infty}^\infty e^{\frac{-1}{2}\left(ax^2+\frac{b}{x^2}\right)}dx$$ where $a,b>0$. And I have the following substitution as a hint: ...
4
votes
1answer
205 views

Evaluate Complex Integral with $\frac{\Gamma(\frac{s}{2})} {\Gamma\big({\beta +1\over 2} - {s\over 2}\big)}$

I am proving this integral: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\beta^{1 \over 2}\,\right)^{s}\ \Gamma\left(\,s \over 2\,\right) \Gamma\left(\,{\beta +1 \over 2} - {s \over ...
4
votes
2answers
57 views

Calculating Triple Integral

I have task : find volume of body limited by surface $(\frac{x}{a})^{2/3} + (\frac{y}{b})^{2/3} + (\frac{z}{c})^{2/3}$ = 1. I know that this task is about triple integral. But i have confused by such ...
4
votes
1answer
156 views

Solving this complicated integral using the Residue Theorem

The following is an integral I am trying to evaluate $$I= \int_{-\infty}^\infty f(s) \, ds = \int_{-\infty}^\infty \frac{\frac{1}{(1- \ \ 2 \pi j s )^{m}}-1}{2\pi j s }\ e^{-2\pi j s \ \theta}\ ds ...
4
votes
2answers
81 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
96 views

How prove this integral $\int_{0}^{1}\frac{(x-1)dx}{(x+1)\ln{x}}=\ln{\frac{\pi}{2}}$ [duplicate]

show that $$I=\int_{0}^{1}\dfrac{(x-1)dx}{(x+1)\ln{x}}=\ln{\left(\dfrac{\pi}{2}\right)}$$ I say this integral background,today I have use this wolf play some function,and Suddenly I found this ...
4
votes
3answers
78 views

Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does given integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we evaluate it's value?
4
votes
1answer
74 views

Evaluation of $\int_{-v/2}^{v/2} \sqrt{1-\left(\frac{t}{7}-\frac{v}{14}\right)^2} \sqrt{1-\left(\frac{t}{7}+\frac{v}{14}\right)^2} dt$

I need to get the value of the following definite integral $v\in \mathbb R^+$ $$\int_{-v/2}^{v/2} \sqrt{1-\left(\frac{t}{7}-\frac{v}{14}\right)^2} \sqrt{1-\left(\frac{t}{7}+\frac{v}{14}\right)^2} ...
4
votes
2answers
60 views

Integration of complicated trignometric function

While solving for power radiation of an dipole antenna, i got stuck at this step during calculation. $\displaystyle \int_0^{\pi}\frac{cos^2(\frac{\pi}{2}cos\theta)}{sin\theta} d\theta$ the methods i ...
4
votes
3answers
152 views

Replace a sum with an integral $\sum\rightarrow \int$

How can one turn a sum to an integral. Example $$\sum_k f(k) \approx N\cdot\int_k dk\, f(k). $$ How do you find the factor $N$? The quantities should be approximately equal. Example form Peskin ...
4
votes
2answers
94 views

Evaluation of $\int \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\ $

How do we find $$\int \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\ $$ NB: It is not mandatory that $ax^2+bx+c$ has only a single root
4
votes
1answer
72 views

Calculus Question: Improper integral $\displaystyle\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$

I am curious about evaluation of the following integral $$\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$$ Is it possible to evaluate it? This not my homework but I will share my attempt. I tried ...
4
votes
4answers
179 views

Integration of some floor functions

Can anyone please answer the following questions ? 1) $\int$ $ \left \lfloor{x}\right \rfloor $ $dx$ 2) $\int$ $ \left \lfloor{\sin(x)}\right \rfloor $ $dx$ 3) $\int_0^2$ $\left ...
4
votes
2answers
97 views

How can you explain implicit differentiation?

So I am taking calculus 1 online from a local college (bad idea, but the only thing that fit my schedule). The professor used the notation $f'(x) =$ for EVERY function up until two weeks ago. All of ...
4
votes
1answer
54 views

Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
4
votes
3answers
178 views

Definite integral $\int_0^{2\pi}(\cos^2(x)+a^2)^{-1}dx$

How do I prove the following? $$ I(a)=\int_0^{2\pi} \frac{\mathrm{d}x}{\cos^2(x)+a^2}=\frac{2\pi}{a\sqrt{a^2+1}}$$
4
votes
2answers
59 views

what is the best way to solve this integral:

have this integral, and looking for the best\quickest way to solve it: $$\int_{-b}^bD\sin\left({\pi ny \over b}\right)\sin\left({\pi n'y \over b}\right)dy$$
4
votes
2answers
199 views

Evaluate $\int_{0}^{\pi}\sin^5{\theta}\cos^2{\theta}\ d\theta$ [duplicate]

I'm trying to find the mass of a spherical object with a given density function, and to do so I must solve this integral $$\int_{0}^{\pi}\sin^5{\theta}\cos^2{\theta}\ d\theta,$$ but no matter which ...
4
votes
2answers
42 views

Proving inequalities about integral approximation

We can state that, with $n$ integer, $$\int_1^n \log x \ \mathrm{dx} \leq \sum_{m = 1}^n \log m$$ because the second is the area of $n$ rectangles with unity base, while the first is "just" the area ...
4
votes
3answers
96 views

Calculate the integral $\int_{0}^{+\infty }[e^{-(\frac{a}{x})^{2}} -e^{-(\frac{b}{x})^{2}}]dx$

Calculate $$\int_{0}^{+\infty }\left[e^{-(\frac{a}{x})^{2}} -e^{-(\frac{b}{x})^{2}}\right]dx,$$ with $0<a<b$ I try to construct a inner parametric integral and change the integration order, but ...
4
votes
1answer
135 views

How to reproduce the Mathematica solution for $\int(\cos x)^{\frac23}dx$?

I entered this integration problem to Mathematica Online Integrator an got a solution I would never have been able to find manually. $$\int\root 3 \of{\cos(x)^2}\,dx=\frac{(-3\cos(x)\root 3 ...