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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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5 votes
2 answers
335 views

Why can't Wolfram Alpha calculate $\int_0^{2\pi}\sqrt{(a-\cos\theta)^2+\sin^2\theta}\ d\theta$?

In this answer to How is the average distance between 2 objects orbiting around a third object calculated? I had to integrate $$\int_0^{2 \pi}\sqrt{(a-\cos \theta)^2 + \sin^2 \theta} \ d\theta.$$ I ...
18 votes
3 answers
923 views

Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$

I'm interested in the following integral: $$\mathcal J(n)=\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx,\tag1$$ where $K(z)$ is the complete elliptic integral of the 1ˢᵗ ...
12 votes
6 answers
599 views

How can I prove $\int_{0}^{1} \frac {x-1}{\log(x) (1+x^3)}dx=\frac {\log3}{2}$

Question:- Prove that $$\int_0^1 \frac {x-1}{\log(x) (1+x^3)} \, dx = \frac {\log(3)}{2}$$ I saw this problem as an comment on a youtube video few hours ago but I don't know how to prove this one as ...
0 votes
0 answers
10 views

Is there a known way to get the average length of a ray bounded by a cone and a box?

I am struggling to come up with a solution to the following problem: Imagine a cone defined by angle $\theta$ that passes through a rectangular prism as is shown below. How would one determine the ...
3 votes
0 answers
188 views

Integral area of $\int_{1}^{e}\frac{3\ln\left(x\right)}{x \sqrt{\ln^{2}\left(x+1\right)}}dx$

Could someone please help me to solve this integral area: $\int_{1}^{e}\frac{3\ln\left(x\right)}{x\cdot\sqrt{\ln^{2}\left(x+1\right)}}dx$ I have no clue how to do it and I tried to do substitution and ...
2 votes
0 answers
55 views
+150

Evaluting $\int_0^\infty K_\nu(ax) K_\nu(bx) \sin(cx) dx$ missing from Erdelyi

There is a known integral, which converges for $\text{Re}(a+b)>0$, $c>0$ and $|\mathrm{Re}(\nu)| < \frac{1}{2}$ such that $$ \int_0^\infty K_\nu(ax) K_\nu(bx) \cos(cx) dx = \frac{\pi^2}{4\...
-4 votes
2 answers
114 views

Integral calculation of$ \int_{1}^{e}\frac{\ln^{2}\left(x\right)}{x\left(1+\ln\left(x\right)\right)}dx$ [closed]

I've been trying to calculate the area of this integral: $$ \int_{1}^{\rm e}\frac{\ln^{2}\left(x\right)}{x\left[1 + \ln\left(x\right)\right]}{\rm d}x $$ and the answer that I get is $\ln\left(2\right) ...
0 votes
0 answers
24 views

How do I take this 2D integral of an exponential function that has a truncation?

I have a 2D distribution that goes like $f(x,y) = exp(-\frac{x^2 +y^2}{v_t^2})$ that I want to truncate at a given radius away from the origin. The truncation line has the form $y(x) = \frac{x^2 - a^...
1 vote
0 answers
28 views

I want integration to turn $x^u$ into $\binom{m-u}{a-u}$

Like we have $$ n! = \int_0^\infty e^{-t}t^n dt $$ and $$ \binom{n}{k}^{-1} = (n+1) \int_0^1 t^{n-k}(1-t)^k dt $$ I would like to have a similar formula for the binomial coefficient $\binom{m-u}{a-u}$....
0 votes
1 answer
16 views

A Point in Sec. 7.8 in Apostol's "Mathematical Analysis ...", 2nd edition: Is $S(P, f) \leq \lvert f(c) \rvert \lVert P \rVert$?

This post is regarding a point made toward the end of Sec. 7.8 in the book Mathematical Analysis - A Modern Approach To Advanced Calculus by Tom M. Apostol, 2nd edition: While trying to bring out the ...
2 votes
1 answer
91 views

Help needed for decay property of the following integral

This integral looks almost like a Fourier-integral, if it wasn't for the limits: $$\int^\infty_{\sqrt{1+1/p^2}}dx\left(\sqrt{\frac{x^2-1/p^2}{x^2-1}}-1\right)e^{-itpx},$$ where $t$ is some positive ...
1 vote
2 answers
915 views

Cases where Simpson's rule has a greater error than Trapezoidal rule?

The error bound formulas for trapezoidal rule and simpson's rule say that: $\begin{array}{l}{\text { Error Bound for the Trapezoid Rule: Suppose that }\left|f^{\prime \prime}(x)\right| \leq k \text ...
3 votes
2 answers
130 views

I need help to evaluate Integral :$I=\int_0^{\pi} \frac{\arctan^2(\sin(x))}{1+\cos(x)}dx$

I need help to evaluate Integral :$$I=\int_0^{\pi} \frac{\arctan^2(\sin(x))}{1+\cos(x)}dx$$ All the tricks I know didn't work. I know that it is my duty to try, but this integration is difficult. I ...
20 votes
1 answer
1k views

Integrating a large product of sines

Recently, I came across the following integral: $$\int_{0}^{2\pi}\sin(x)\sin(2x)\sin(3x)\sin(4x)~\mathrm dx=\frac{\pi}{4}$$ which can be easily solved by some trigonometry. But when trying to find a ...
-1 votes
0 answers
32 views

What other ways can we derive a Riemann sum by thought experiment?

I was thinking about the traditional thought experiment that is used to derive the Riemann sums. I realized that Riemann sums can be derived with two other thought experiments (see 2 and 3). Are there ...
-2 votes
1 answer
62 views

Integration of a very hard expression [closed]

I came across this integration problem in my module but I am unable to crack it. Here it is-- Find in terms of 'a'. $$\int_0^\infty {x^a}e^{-x} dx$$ I have tried substitution and integration by parts ...
1 vote
0 answers
33 views

Check the Integration process for this integration: $\int_0^s \exp\left(-ax+\frac bx\right)~dx$

I am trying to integrate $\int_0^s \exp\left(-ax+\frac bx\right)~dx$. I proceed in this way (according to Primitive of $ \exp(-a \times x- \frac bx)$): $=\int_0^s \exp(-ax-(\frac {(-b)}{x})~)~dx$ $=s\...
0 votes
0 answers
68 views

Calculate $\lim_{n\to \infty}\int_0^1 \cos^nx \, dx$ without using newton leibniz [duplicate]

I'm asked to calculate: $$ \lim_{n\to \infty} \int_0^1 \cos^n x \, dx, $$ So far I have reached this: $$ \lim_{n\to \infty} \int_0^1 \cos^n x \, dx = \lim_{n\to \infty}\frac{n-1}{n} \int_0^1 \cos^{...
2 votes
3 answers
162 views

Integrate and find closed form for $\int_0^\infty\frac{\sin x^n}{n\pi}dx$

I’m working on a practice exam for my analysis class, and I was asked to find a general form for $$\int_0^\infty\frac{\sin x^n}{n\pi}dx$$ When I first looked at this, my mind instantly went to the ...
3 votes
1 answer
83 views

Show that $\int_0^1K^2(k)dk=\frac12\int_0^1K'^2(k)dk$

By switching integrals in double integral, I showed that $$\int_0^1K(k)dk=\int_0^1K'(k)dk=2G$$ where $K(k)$ is complete elliptic integral of the first kind and $K'(k)=K(\sqrt{1-k^2})$ is its ...
0 votes
0 answers
51 views

floor of a non elementary integral [closed]

if $\alpha$ is the greatest integer lesser or equal to $\beta$ where $\beta = \int_{2}^{3} \frac{x}{\ln x} \,dx $ $\alpha$ = ? My attempt: What I tried doing is study the function $f(x) =\frac{x}{\ln ...
1 vote
1 answer
36 views

Elliptic integral singular expansion

The question. Consider the Elliptic Integral $$ F(x;k)=\int_0^x \frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}}.\tag{1}\label{1} $$ I am interested in the singular series expansion of $F(1;k)$ about $k=1$. I was ...
6 votes
2 answers
168 views

How to evaluate this integral like the Ahmed's integral?

How to evaluate that : $$ \int_0^1 \frac{\tan^{-1}\sqrt{2-x^2}}{1+x^2}dx =\frac{7}{96}\pi^2 $$ I'm trying to convert it into a double integral. Notice : $$ \left[\tan^{-1}\left(\frac{y}{\sqrt{2-x^2}}\...
0 votes
0 answers
63 views

Laplace transform of exponential functions with derivatives.

I have been trying to calculate the Laplace transform of these troublesome exponential functions: Having $\alpha \in \mathbb{R^+}$ 1.$\mathcal{L}\left\{e^{n \alpha t}\frac{f(t)}{t^2} \right\},n \in \...
0 votes
1 answer
55 views

"De-singularize" an improper integral

Consider the improper integral $$ \int_0^\pi \frac{dx}{\sqrt{(\cos(x)+1+a)(\cos(x)+1)}}, $$ for some $a>0$. Does anyone know of a change of variables substitution that would make this integral ...
0 votes
2 answers
117 views

Is there a way to calculate $I_{p,q}=\int_0^1 t^p(1-t)^q dt\text{ with } (p,q) \in \mathbb{N^2}$ without using the gamma function? [closed]

Given $I_{p,q}=\int_0^1 t^p(1-t)^q dt\quad \mbox{with}\quad (p,q) \in \mathbb{N^2}$, I tried by calculating $I_{0,0}$ and $I_{1,1}$ and so on until I got $I_{p,q}=\frac{p!\, q!}{\left(p + q\right)!}.$
-1 votes
2 answers
32 views

How do you procede demonstrate that an integral is dervivable. [closed]

Let's take ${f} : [a; b] \rightarrow \mathbb{R}$, continuous on$[a; b]$ And $\Phi : [a; b] \rightarrow \mathbb{R}$ where $\Phi(x) = \int_{a}^{x}{f}(t)dt$ In the démonstration of the Fundamental ...
0 votes
0 answers
76 views

Question about $\int_{\frac{\pi }{2}}^{\frac{3 \pi }{2}} \tan (x) \, dx$

Let's consider: $$\int_{\frac{\pi }{2}}^{\frac{3 \pi }{2}} \tan (x) \, dx$$ The graph of the function within the integration interval looks like this: Sorry if the answers are obvious, but my ...
25 votes
4 answers
3k views

Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$$ My first thought was to use the beta function, but it's hard to ...
19 votes
5 answers
1k views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...
13 votes
2 answers
364 views

Prove known closed form for $\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx$

I know that the following identity is correct, but I would love to see a derivation: $$\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx=\frac{\sqrt{6}}{32\pi^3}\Gamma\left(\frac{1}{24}\right)\...
8 votes
3 answers
265 views

Evaluate $\int_{-1}^{1}\sin^{-1}\left(x\right)\tan^{-1}\left(x\right)\mathop{\mathrm{d}x}$

I tried integration by parts $$\begin{aligned} \int_{-1}^{1}\sin^{-1}\left(x\right)\tan^{-1}\left(x\right)\mathop{\mathrm{d}x}&=2\int_{0}^{1}\sin^{-1}\left(x\right)\tan^{-1}\left(x\right)\mathop{\...
2 votes
0 answers
54 views

Finding Volume of Revolution Given by $y = \sin x$

The question given is to find the volume of revolution generated by the graph of $y = \sin x$ on the interval $[0, \pi]$. The way I attempted was to form the sums of cylindrical segments given by $\...
1 vote
0 answers
30 views

Getting suspiscious wrong results when evaluating definite integral over spatial frequency

I'm currently facing a problem evaluating an integral from this paper (page 36 ff.) $$ \int_{all \,\xi} |H_{post}(\xi)H_{eye}(\xi)|^2d\xi $$ where $\xi$ is a spatial frequency and $$ H_{post}(\xi) = e^...
2 votes
0 answers
50 views

Volume generated by revolving $\sin x \cos x$ around x-axis

Question: find the volume generated when the region bounded by $y = \sin x \cos x, 0\le x \le \frac{\pi}{2}$, is revolved about the x-axis. This question appeared quite tricky, and the book that ...
2 votes
2 answers
113 views

How to evaluate $\int_0^{\frac{\pi}{2}}(\cos t)^{\frac{5}{2}}{\rm d}t$?

Let $\sqrt{\cos t}=:u$,then $t=\arccos u^2$,${\rm d}t=-\dfrac{2u}{\sqrt{1-u^4}}{\rm d}u$. Therefore \begin{align*} \int_0^{\frac{\pi}{2}}(\cos t)^{\frac{5}{2}}{\rm d}t&=2\int_0^{1}\frac{u^6}{\sqrt{...
1 vote
2 answers
212 views

How to evaluate $\int_{0}^{\frac{\pi}{2}} \frac{1}{(1 + \sin^4(x))\sqrt{1+\sin^2(x)}}\, dx$

How would you calculate the integral without using identities involving integrals/hyper geometric functions? \begin{align} \int_{0}^{\frac{\pi}{2}} \frac{1}{(1 + \sin^4(x))\sqrt{1+\sin^2(x)}}\, dx \...
1 vote
1 answer
71 views

Bessel function of first and second kind as solution of an integral

Needing to solve this definite integral $$ \int_{0}^{\infty} \frac{1}{\sqrt{u}}\exp\left[{-\frac{(t-u)^2}{2}} \right] du$$ where $t$ is real, someone make me notice that, although it seems that there ...
-4 votes
1 answer
61 views

if f is a continuous function such that $\int_1^8 f(x) \,dx= 9$, then $\int_1^2 t^2f(t) \,dt$ [closed]

The answers are: =9 =3 =27 Provided data is not enough I think it can be solved integrating by parts, but i dont find the correct way of doing it. Thanks in advance!!
5 votes
1 answer
143 views

Closed form of $ \int_0^\infty dx r J_1(r x) \left [ J_0(x) \right ]^Q $

I'm looking for a closed form for the following definite integral $$ I(r,Q) := r \int_0^\infty dx J_1(r x) \left [ J_0(x) \right ]^Q $$ where $r$ is a positive real, $Q$ is a positive integer and $J_a$...
0 votes
0 answers
53 views

fourier transform $1/\sqrt{1-(x+ia)^2}$

I am trying to evaluate the Fourier Transform $$ f(z)=\int_{-\infty}^\infty dx \exp(-ixz)\frac{1}{\sqrt{1-(x+ia)^2}}\quad \mbox{where}\ a > 0 $$ Does anyone have ...
0 votes
1 answer
97 views

Exponential Cosine Gaussian Integral [closed]

through the course of some computations I've come across this integral: $$ \int_{-\infty}^{\infty} e^{a \cos(t) - b t^2} dt \qquad a,b \in \mathbb{R}, \quad b>0 $$ which is by far the simplest (...
10 votes
1 answer
643 views

Closed form for integral of inverse hyperbolic function in terms of ${_4F_3}$

While attempting to evaluate the integral $\int_{0}^{\frac{\pi}{2}}\sinh^{-1}{\left(\sqrt{\sin{x}}\right)}\,\mathrm{d}x$, I stumbled upon the following representation for a related integral in terms ...
6 votes
4 answers
260 views

A difficult integral $I=\int_0^1\sqrt{1+\sqrt{1-x^2}}\frac{dx}{1+x^2}$

How to prove $$I=\int_0^1\sqrt{1+\sqrt{1-x^2}}\frac{dx}{1+x^2}=\sqrt{\sqrt{2}+1}\arctan\sqrt{\sqrt{2}+1}-\frac{1}{2}\sqrt{\sqrt{2}-1}\ln(1+\sqrt{2}+\sqrt{2+2\sqrt{2}})$$ $$ I=\int_0^{\pi/4}\sqrt{1+\...
1 vote
0 answers
29 views

Show that $H_n$ represents a bilinear form

Recently, I had a linear algebra quiz which included the following extra problem (which I didn't have enough time to solve): For all $n\geq 1$, let $H_n\in M_n(\mathbb{R})$ be the matrix whose entry $(...
10 votes
5 answers
457 views

Prove $\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2} \right)$

Wolfram Alpha evaluates this integral numerically as $$\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=0.379064 \dots$$ Its value is apparently $$\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}...
1 vote
2 answers
3k views

Find the volume of solid generated by rotating this sector?

Find the volume of the solid generated by revolving the plane region bounded from upward by circle $x^2 + y^2 = 4$ and from downward by two straight lines $y = x$ and $y = -x$ around $x-axis$ So as I ...
0 votes
2 answers
144 views

Evaluate the following improper integral $\int_0^\infty (\ln(1-e^{-x}))^ce^{-ax}x^bdx, \quad a>0, b,c \in \mathbb{N}_0$

Evaluate the improper integral $$ \int_{0}^{\infty}\ln^{\large c}\left(1 - {\rm e}^{-x}\right) {\rm e}^{-ax}x^{b}\,{\rm d}x, \quad a > 0,\ b,c \in \mathbb{N}_{0} $$ Is it possible to find closed ...
0 votes
2 answers
106 views

How to integrate $\int\frac{x(1-x)}{(1+x)}dx$

Q)How to integrate $$\int\frac{x(1-x)}{(1+x)}dx$$ My Approach : I know how to integrate these types of integrals. We have to substitute $(1+x)=u$. Then $dx=du$. Therefore the integral will become : $$\...
0 votes
1 answer
93 views

Showing $\int_0^\pi\ln\left|1+ae^{-jx}\right|\,dx=0$, where $|a|\leq1$, and $j$ is the imaginary unit [duplicate]

It is said that the following integral equals 0 from some ADC circuit design book. I tried to prove it but did not work out. $$ \int_{0}^{\pi} \ln\left( \left| 1 + a e^{-jx} \right| \right) \, dx = 0 \...

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