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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3answers
803 views

integrating $\int \sqrt{2-2\cos(x)} \, dx$

So i am having some trouble getting the solution to the integral: $\int \sqrt{2-2\cos(x)} \, dx$ i made my first substitution $u = 2-2\cos(x)$ $u' = 2\sin(x) \, dx$ then... $\int ...
4
votes
3answers
1k views

Finding volume using triple integrals.

Use a triple integral to find the volume of the solid: The solid enclosed by the cylinder $$x^2+y^2=9$$ and the planes $$y+z=5$$ and $$z=1$$ This is how I started solving the problem, but the way I ...
4
votes
3answers
89 views

Evaluate $\int_0^R \frac{r^2}{(1+r^2)^2}dr.$

I am trying to evaluate the following integral: $$\int_0^R \frac{r^2}{(1+r^2)^2}dr.$$ I might substitute $u=r^2$, but I don't find $du$ anywhere. Obviously the integral should be bounded on $R\in ...
4
votes
3answers
608 views

Integrating Modified Bessel function of the second kind?

I need to compute the following integral: $$\int_0^\infty\;\;K_0\left(\sqrt{a(k^2+b)}\right)dk$$ where $a>0$ and $b>0$. I have tried several substitutions and played around a lot in ...
4
votes
2answers
622 views

How to integrate $\int{e^{\tan^2{x}}\sin(4x)}dx$

The question says it all. How do I even go about integrating this integral. I'll appreciate some help. Thanks in advance. $$\int{e^{\tan^2{x}}\sin(4x)}dx$$
4
votes
1answer
612 views

Integrating $\exp (\exp (x))$

How am I able to integrate $e^{e^x}$? $$\int e^{e^x}dx$$ Am I suppose to use $u$ substitution? But what should I let $x$ be? And what should $dx$ be? Thanks for the help!
4
votes
5answers
442 views

Integrate $\int_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$.

Integrate $\displaystyle \int \limits_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$. I tried substituting $t=\cos x$, and then integrate with integration by parts. It got all messy... Thanks in advance for any ...
4
votes
2answers
436 views

Double integral application

I need to determine $$\int_{0}^{1} \int_{-\sqrt{x}}^{\sqrt{x}}\frac{1}{1-y}dydx$$ I integrate in terms of the y component and obtained: $$\int_{0}^{1}\ln(\frac{1+\sqrt{x}}{1-\sqrt{x}})dx$$ Can ...
4
votes
1answer
349 views

Integration of an integral part of x?

If $ f(x)={\left\lfloor x^2\right\rfloor -\left\lfloor x\right\rfloor ^2}$,where ${\left\lfloor x\right\rfloor }$ denotes the greatest integer $\le x$ then $\int_1^2 f(x)dx?$ please give some hint. ...
4
votes
2answers
378 views

$\lim_{n\to\infty}\int_0^n\left(1-\frac{x}{n}\right)^n\text{e}^{\frac{x}{2}}\text{d}x$ Evaluating this limit

How to evaluate : $$\lim_{n\to\infty}\int_0^n\left(1-\frac{x}{n}\right)^n\text{e}^{\frac{x}{2}}\text{d}x$$
4
votes
3answers
167 views

Help in manipulating Integrals

I try to express : $\displaystyle 1+2\sum _{ k=1 }^n \cos(2k\theta ) $ as : $\dfrac { \sin\left( \theta +2\theta n \right) }{ \sin\left( \theta \right) } $ I tried to use the exponential function ...
4
votes
2answers
2k views

Riemann sum of $\sin(x)$

I would like to calculate the Riemann sum of $\sin(x)$. Fun starts here: $$R = \frac{\pi}{n} \sum_{j=1}^n \sin\left(\frac{\pi}{n}\cdot j\right)$$ What would be the simplest way to calculate the sum ...
4
votes
1answer
1k views

integral from 0 to $2\pi$ of $|\cos x|\operatorname{d}x$ not integrating as I'd expect

I drew a rough sketch of $|\cos x|$ and would guess the correct answer to this integral is $4$ because I know the area under the curve of $\cos x$ from $0$ to $\pi/2$ is $1$, and there are $4$ such ...
4
votes
5answers
183 views

Order of solving definite integrals

I've been coming across several definite integrals in my homework where the solving order is flipped, and am unsure why. Currently, I'm working on calculating the area between both intersecting and ...
4
votes
2answers
459 views

Questions about Fubini's theorem

I was wondering what theorem(s) makes possible exchanging the order of Lebesgue integrals, for instance, in the following example: $$\int_0^1 \int_0^x \quad 1 \quad dy dx = \int_0^1 \int_y^1 ...
4
votes
1answer
329 views

How we get the area by subtracting two end points of a function in Integration?

For example consider the following integration: f(x) = x^3 [from 1 to 3] $\int_{1}^{3}x^{3}dx$ when we subtract: {(3^4)/4}-{(1^4)/4} why we get the result? I meant to say, how we get the area by ...
4
votes
3answers
142 views

Closed-form of $\int_0^1 \operatorname{Li}_3\left(1-x^2\right) dx$

By using dilogarithm functional equations we can show that $$ \int_0^1 \operatorname{Li}_2\left(1-x^2\right)\,dx = \frac{\pi^2}{2}-4, $$ where $\operatorname{Li}_2$ is the dilogarithm function. Could ...
4
votes
4answers
185 views

How to prove $\lim\limits_{x\to\infty}\int_a^bf(t)\sin(xt)\,dt=0$

I need help to prove: Suppose $f\in C$. Prove that $$ \lim\limits_{x\to\infty}\int_a^bf(t)\sin(xt)\,dt=0 $$ My idea is to use substitute of $xt=u$ and prove $$ ...
4
votes
5answers
137 views

How to evaluate $\lim _{n\to \infty }\:\int _{1/(n+1)}^{1/n}\:\frac{\sin\left(x\right)}{x^3}\:dx$?

We have to evaluate the following limit: $$\lim _{n\to \infty }\:\int _{\frac{1}{n+1}}^{\frac{1}{n}}\:\frac{\sin\left(x\right)}{x^3}\:dx,\:n\in \mathbb{N}$$ First step I wrote that $\int ...
4
votes
2answers
150 views

a way to integrate: $\int (\sqrt{x} +3)/(2+ x^ \frac{1}{3}) dx$

Im looking for a way to integrate: $$ \int \frac{ \sqrt{x} +3}{2+ x^ \frac{1}{3}} dx $$ that would make it efficient and not too difficult... Any suggestions?
4
votes
3answers
125 views

Find $ \int \frac {1-x^2}{1+3x^2+x^4} \, \mathrm{d}x $

Today, the CalcBee sample problems got released. The following problem was my creation and I wanted to see how many solutions people can come up with. The result is very beautiful and I thought it ...
4
votes
1answer
78 views

Evaluating the integral $\int_{-1}^{1}\frac{x^2\,dx}{e^x+1}$

Can this integral be evaluated properly by using highschool integration skills ? $$\int_{-1}^{1}\dfrac{x^2}{e^x+1}\,dx$$ (Original image at http://i.stack.imgur.com/AqOsX.png) Judging from what I ...
4
votes
5answers
214 views

Integral of $1/[(1+x^2)\sqrt{1+x^2}]$

I try to get back on track with the integration. I would like to solve $$ \int_0^1 \frac{dx}{(1+x^2)\sqrt{1+x^2}}.$$ There are my way to try to solve it (that I don't find the right solution) and an ...
4
votes
2answers
110 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
4
votes
3answers
117 views

Proving $\int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1$.

By testing in maple I found that $$ \int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1 $$ Does there exists a proof for this? I tried rewriting it as an series but no luck ...
4
votes
5answers
186 views

Evaluate $\int \sqrt{1-x^2}\,dx$

I have a question to calculate the indefinite integral: $$\int \sqrt{1-x^2} dx $$ using trigonometric substitution. Using the substitution $ u=\sin x $ and $du =\cos x\,dx $, the integral becomes: ...
4
votes
4answers
237 views

How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$?

I have no idea how to start, it looks like integration by parts won't work. $$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$ If someone could shed some light on this I'd be very thankful.
4
votes
2answers
103 views

Calculate $\int_{0}^{1}\dfrac{x^{2n}}{\sqrt{1-x^2}}\mathrm{d}x$

For integer $n\ge0$, Calculate: $$\int_{0}^{1}\dfrac{x^{2n}}{\sqrt{1-x^2}}\mathrm{d}x.$$ I would like to get suggestions on how to calculate it? Should I expand $(1-x^2)^{-1/2}$ as a series? Thanks. ...
4
votes
3answers
291 views

Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$

Hi I am trying to prove this $$ I:=\int_0^{\pi/4}\log\left(\tan\left(x\right)\right)\, \frac{\cos\left(2x\right)}{1+\alpha^{2}\sin^{2}\left(2x\right)}\,{\rm d}x ...
4
votes
2answers
237 views

What is the integral of n-th root of tan x?

What will be the value of $ \int\sqrt[n]{\tan x},dx $ ? I have solved the cases for n=2 and n=3 but can't see how I can generalize it.
4
votes
4answers
170 views

Fast calculation for $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$

I want to show that $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$, but is there a faster method than finding the contour and doing all computations? Otherwise my idea is to do the substitution $x=e^t$, ...
4
votes
4answers
145 views

Compute the definite integral

Find $$\int_0^{\pi}\frac{x}{1+\cos^2(x)}dx$$ I tried letting $u=\tan(\frac{x}{2})$ but could not make it work. A few other trig substitutions failed as well. I noticed the integrand is odd but could ...
4
votes
4answers
123 views

Inequality $\left(\int_0^1 f(x)dx\right) \left(\int_0^1 \frac{1}{f(y)} dy\right) \ge 1$

Let $f$ be a positive continuous function defined on a closed interval $[0,1]$, then it is true that: $$\left(\int_0^1 f(x)dx\right) \left(\int_0^1 \frac{1}{f(y)} dy\right) \ge 1$$ I tried to show ...
4
votes
3answers
131 views

Find value of integral: $I=\int_0^{2\pi}\frac{dx}{(2+\cos x)^2}$

Find value of integral: $$I_1=\int_0^{2\pi}\frac{dx}{(2+\cos x)^2}$$ and $$I_2=\int_0^{2\pi}\frac{dx}{(2+\sin x)^2}$$ I don't know how, i need a solution, please
4
votes
4answers
168 views

How find this integral $\int\frac{1}{1+\sqrt{x}+\sqrt{x+1}}dx$

Question: Find the integral $$I=\int\dfrac{1}{1+\sqrt{x}+\sqrt{x+1}}dx$$ my solution: let $\sqrt{x}+\sqrt{x+1}=t\tag{1}$ then $$t(\sqrt{x+1}-\sqrt{x})=1$$ $$\Longrightarrow ...
4
votes
3answers
264 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
4
votes
2answers
213 views

Definite Integral $\int_{-\pi}^{\pi}\frac{e^{\sin(x)+\cos(x)}\cos(\sin(x))}{e^{\sin(x)}+e^x}dx$

How can I find the value of this following definite integral? $$\int_{-\pi}^{\pi}\frac{e^{\sin(x)+\cos(x)}\cos(\sin(x))}{e^{\sin(x)}+e^x}dx$$ Mathematica says it's value is $\pi$, but I don't ...
4
votes
3answers
848 views

How to integrate it $\int_{0}^{b}\ln(x+\sqrt{x^2+1})dx=?$

I would appreciate if somebody could help me with the following problem: Q:How to integrate it $$\int_{0}^{b} \ln(x+\sqrt{x^2+1})dx=?(b>0)$$
4
votes
2answers
610 views

Arc length $y = \frac{1}{4} x^2 - \frac{1}{2} \ln x$

$$y = \frac{1}{4} x^2 - \frac{1}{2} \ln x$$ $$\int_1^{2e} \sqrt{1 + (y')^2}$$ $$y' = \frac{x}{2} - \frac{1}{2x}$$ $$y' = \frac{2x^2-1}{2x}$$ $$\left(\frac{2x^2-1}{2x}\right)^2$$ ...
4
votes
4answers
319 views

Finding indefinite integral 2

$$\int\frac{\sqrt{a+x}}{\sqrt{a}+\sqrt{x}}\, dx$$ A few days ago I asked a similar (looking) question where all the pluses here were minuses. It could be much more easily manipulated than this one ...
4
votes
6answers
857 views

Computing $\int \frac{6x}{\sqrt{x^2+4x+8}} dx$

I am trying to compute an indefinite integral. Thanks! $$\int \frac{6x}{\sqrt{x^2+4x+8}} dx.$$
4
votes
3answers
142 views

How to solve the improper integral $\int_{-\infty}^{\infty} \frac{x^2}{x^6+9}dx$ (possible trig substitution)

$$\int_{-\infty}^{\infty} \frac{x^2}{x^6+9}dx$$ I'm a bit puzzled as how to go about solving this integral. I can see that it isn't undefined on -infinity to infinity. But I just need maybe a hint on ...
4
votes
3answers
2k views

Integrate: $\int x(\arctan x)^{2}dx$

I'm not sure how to start I think we have to use integration by parts $$\int x(\arctan x)^{2}dx$$
4
votes
2answers
10k views

Can a limit of an integral be moved inside the integral?

After coming across this question: How to verify this limit, I have the following question: When taking the limit of an integral, is it valid to move the limit inside the integral, providing the ...
4
votes
4answers
4k views

Why is the differential solid angle have a $\sin\theta$ term in integration in spherical coordinates?

When you integrate in spherical coordinates, the differential element isn't just $ d\theta d\phi $. No. It's $\sin\theta d\theta d\phi$, where $\theta$ is the inclination angle and $\phi$ is the ...
4
votes
3answers
3k views

How to integrate $\int e^{-t^{2}} \space dt $ using introductory calculus methods

Earlier today I stumbled across this when I was doing some practice questions for a physics course: $$\int e^{-t^2} \space dt $$ To expand, the limits of integration were something like $1$ and $4$ ...
4
votes
2answers
200 views

Is there a formula for solving integrals of this form?

I was wondering if there was a substitution formula to solve integrals of this form: $\int f(g(x))g''(x)dx$
4
votes
4answers
241 views

Integral question: $\displaystyle\int \frac{x^{n-2}}{(1 + x)^n} {\rm d}x$

How would one integrate the following? $$\int \frac{x^{n-2}}{(1 + x)^n} {\rm d}x~$$ where $n$ is a positive integer.
4
votes
1answer
542 views

Computing the integral $\int_{0}^{\pi/2} x\,\arccos\left(x^{2}\right)\,{\rm d}x$

How can the following integral be computed $$ \int_{0}^{\pi/2} x\,\arccos\left(x^{2}\right)\,{\rm d}x\quad{\large ?} $$
4
votes
2answers
93 views

Calculating $\int_0^{\infty} \frac{\log^2(1 - e^{-x})\:x^5}{e^x - 1} \: dx $ [duplicate]

I am having trouble calculating the following improper integral: $$\displaystyle \int\limits_0^{\infty} \frac{\log^2(1 - e^{-x})x^5}{e^x - 1} \, dx $$ Can someone give me a way that I can calculate ...