Questions about the evaluation of specific definite integrals.

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2
votes
2answers
84 views

Partial-Fraction Decomposition

So I was doing some integrals and ran across this one: $$\int{\frac{3x+1}{x^2+4x+4}}dx=\int{\frac{3x+1}{(x+2)(x+2)}}dx$$ Of course, I started decomposing the fraction and immediately realized it ...
4
votes
2answers
67 views

How to find the definite integral $\int_0^\infty \frac{x}{\sinh ax}\;dx$

I'm trying to prove that $$I:= \int_0^\infty \frac{x}{\sinh(ax)} dx = \frac{\pi^2}{4a^2}$$ Attempt: $$\sinh (ax) = \frac{1}{2}(e^{ax}-e^{-ax}) = \frac{1}{2}e^{-ax}(e^{2ax}-1)$$ Now I have ...
-2
votes
1answer
38 views

Limit of integral with continuous function [on hold]

Compute the following limit: \begin{equation} \lim_{n \to \infty} \int_0^1 \sin (n \pi x)g(x)dx, \end{equation} where $g$ is a continuous funtion.
0
votes
0answers
10 views

Finding the conditions for $\int_a^b P_X(Q,X)dQ = P_X(Q,X) b$ [on hold]

I'm trying to find the conditions on the function P such that $\int_a^b P_X(Q,X)dQ = P_X(Q,X)* b$. I don't know where to start. I tried to use integration by parts, but it didn't help.
4
votes
0answers
66 views

Other integral related to Ahmed's integral [on hold]

I have a doubt regarding the evaluation of the following integral : $$ \int_0^\frac{1}{\sqrt{5}} \frac{\tan^{-1}\left({\sqrt{(1 + x^2)/2}}\right)} {(1 + 3x^2)\sqrt{1 + x^2}}\,du = ...
4
votes
0answers
58 views

How to integrate $\frac{x^{2}\log {\sin x}}{1+x^{6}}$

I recently stumbled upon a question $$\int_0^{\infty}\frac{x^{m-1}\log^{a}x}{1+x^n}dx$$ I was able to evaluate it,but I am curious if there exists a closed form for, ...
1
vote
1answer
17 views

Error esimate given the midpoint estimation of an integral.

I calculated an approximate integral $\left (\int_0^{3.2} f(x)dx \right )$ using midpoints and given data from a table, and got $23.44$. The second derivative of $x$ is said to be between $-4$ and $1$ ...
1
vote
1answer
81 views

Proving that $ \int_{0}^{\pi/2} \frac{\mathrm{d}{x}}{\sqrt{a^{2} {\cos^{2}}(x) + b^{2} {\sin^{2}}(x)}} = \frac{\pi}{2 \cdot \text{AGM}(a,b)} $.

I know Neumann’s solution of this famous definite integral that is totally based on substitution, but is there any solution using complex analysis? Assuming that $ a > b $, show that $$ ...
1
vote
0answers
16 views

If $X \sim N(\mu,\sigma^2)$, then $\int^t_sxf(x)dx=\sigma [f(s)-f(t)]+ \mu [F(t)-F(s)] $?

Here is my work, kindly let me know if this is correct: \begin{align*}\int^t_sxf(x)dx=&\int^{\frac{t-\mu}{\sigma}}_{\frac{s-\mu}{\sigma}}(\sigma z+\mu)\frac{\phi(z)}{\sigma}\sigma dz \\=& ...
1
vote
2answers
36 views

Evaluate the following spherical coordinate triple integral

Evaluate the following \begin{align*} \int_{0}^{\pi} \int_{0}^{\pi/3} \int_{\sec \phi}^{2} 5\rho^2 \sin(\phi) \ d\rho \ d\phi \ d\theta \end{align*} Attempt at solution: We have \begin{align*} 5 ...
0
votes
2answers
52 views

Integration dilemma

I want to compute the value of $$\int_0^{2\pi}\dfrac{4ie^{it}}{4e^{it}-3}\,\mathrm{d}t$$ Using the Maple software, the answer is $2\pi i$. However, when I worked it out myself, I got $0$ instead. ...
0
votes
1answer
16 views

Changing Limits on an Integral

How do I show $$ \int_{t_0}^t \int_{t_0}^t H(t')H(t'') \, dt'' \, dt'= \int_{t_0}^t \int_{t_0}^{t'} [H(t')H(t'') + H(t'')H(t') ]\,dt''\,dt' ?$$ Starting with $ \int_{t_0}^t \int_{t_0}^t H(t')H(t'') ...
3
votes
1answer
54 views

How to integrate $\int \limits_{0}^{\infty} \frac{e^{-(t+\frac{1}{t})}}{\sqrt t} dt$

This is a problem given in my homework . I have to find the integral$$\int \limits_{0}^{\infty} \frac{e^{-(t+\frac{1}{t})}}{\sqrt t}dt$$ I am trying to use integral representation of the gamma ...
1
vote
2answers
97 views

How to integrate $\frac{1}{2+\sin x}$ [on hold]

$\displaystyle\int_0^{\pi/2}\frac{1}{2+\sin x}\,dx$ Thank you in advance.
0
votes
0answers
15 views

Volumes Integration Question help [on hold]

So basically with this question I am kinda stuck and don't really know where to begin as the question is really confusing me. Help would be much appreciated and I would totally forever be thankful to ...
0
votes
2answers
33 views

Volume - Integration Question Help

So basically, with this question I have answered parts a) and b), but I am stuck in relation to part c). I am not exactly sure what the borders of my integration should be, but I figured with the ...
1
vote
2answers
53 views

Show that $f$ is identically zero

Suppose $f:[0,1]\to\mathbb R$ is a continuous function satisfying $$|f(x)|\leq\int_0^xf(t)dt$$ for all $x\in[0,1]$. Show that $f$ is identically zero. I note that $f(0)=0$ trivially. Then how ...
4
votes
1answer
97 views

A definite integral related to Ahmed's integral

Further work on a recent question which has just been answered leads me to make this conjecture: $$ \int_0^1 \frac{\tan^{-1}\left(\frac{1}{\sqrt{2 + x^2}}\right)} {(1 + x^2)\sqrt{2 + x^2}}\,du = ...
0
votes
1answer
27 views

Integrate characteristic function ordering n variables

Given $n$ variables, consider the characteristic function $\mathbb{1}_{x_1<\ldots<x_n}:[0,1]^n\rightarrow\{0,1\}$ where ...
0
votes
0answers
17 views

Alternative way to finding this limit

Statement: Let $f(a,x) = \frac{1}{|x-a|+3}$ where $a,x$ are real numbers and $g:[0,3] \to \mathbb R $ be a continuous function. Calculate $$\lim_{a \to \infty} \int^3_0 f(a,x)g(x) dx$$ My method was ...
0
votes
0answers
18 views

What are some examples of asymptotic expansions of integrals displaying the Stokes phenomenon?

With the term Stokes phenomenon we refer to how the asymptotic behaviour of a function can differ in different regions of the complex plane. What are some examples of asymptotic expansions of ...
0
votes
2answers
59 views

How do I evaluate the integral of the Dirichlet function on $[0,1]$?

Define $f(x) = \left\{ \begin{array}{ll} 1 & \mbox{if } x \in \mathbb{Q} \\ 0 & \mbox{if } \notin \mathbb{Q} \end{array} \right.$ How to evaluate $\int_0^1 f(x)\,dx$ ? I have no ...
-1
votes
1answer
29 views

Normalizing a Definite Integral

Say I have a function $f(x)$ that fulfills the following normalization condition: $\frac{1}{360}\int^{360}_0f(x)dx = 1$ Now say I have a new function $g(x)$ that I want to integrate over a different ...
2
votes
0answers
45 views

Multiple Integral Puzzle [duplicate]

$$I=\int_1^2\int_1^2\int_1^2\int_1^2 {{x_1+x_2+x_3-x_4}\over{x_1+x_2+x_3+x_4}} \,dx_1\,dx_2\,dx_3\,dx_4$$ $I=1/2$ or $1/3$ or $1/4$ or $1$ ? [from ISI-Kolkatta Sample papers] I know that there ...
0
votes
1answer
40 views

Algebraic Properties of the Integral

Prove that $$\frac{1}{3\sqrt{2}} \leq \int_0^1 \frac{x^2}{\sqrt{1+x^2}}\space dx \space \leq \frac{1}{3}$$ Use: If $f_1(x)$ and $f_2(x)$ are integrable on $[a, b]$ and $f_1\leq f_2$ then $$\int_a^b ...
0
votes
0answers
21 views

evaluating a probability on a multivariate normal

Let $(X_1,X_2,X_3,X_4)$ be zero mean jointly normal with covariance $\Sigma$. I would then be thankful if you help me with hints as how to evaluate $$P(X_1^4X_2^2-X_3^4 X_4^2\leq 0).$$ This should ...
0
votes
0answers
18 views

Integrable Functions on a Partition P

Kindly help me with this problem.. ( The underlined statements ).. Please help !!
1
vote
2answers
50 views

Integrate $\int_0^{\pi/2}\cos^8(x)dx$ [on hold]

I'm usually pretty good with definite integrals, but this one's got me completely lost. Any help is appreciated! (And the sooner the better, please!) $$\int_0^{\pi/2}\cos^{8}x\,dx.$$
0
votes
1answer
18 views

Proof of non-integrability for Dirichlet like function

Let $f:[0,1]\rightarrow\mathbb{R}$ be a function which has the value $$f(x)=\left\{ \begin{array}{l l} x^2 & \quad ,x\in\mathbb{Q}\\ x^3 & \quad ,x\notin\mathbb{Q} \end{array} ...
0
votes
0answers
32 views

Analytic solution to diffuclt integral

I'm looking for an analytic solution to the following integral. For $\epsilon > 0$ very small and $a, b, A, B, C$ all reals satisfying $a^2 - b^2 = A^2 - B^2 - C^2$, I'm looking for an analytic ...
0
votes
1answer
10 views

surface area when the curve is revolved about the x-axis

find the surface area when the curve is revolved about the x-axis Steps so far: $y=\sqrt{4x+3}$ on $[0,3]$ $$S=\int_{a}^{b} 2π \left(f(x)\right)\sqrt{1+f'(x)^2} $$ $$S=\int_{a}^{b} 2π ...
3
votes
0answers
28 views

How does the Stokes phenomenon appear in the asymptotic expansion of $\int_0^\infty \frac{e^{-zt}}{1+t^4} dt$ for $z \to \infty$?

Consider the asymptotic $z \to \infty$ behaviour of the function $$ \tag 1 I_1(z) \equiv \int_0^\infty \frac{e^{-zt}}{1+t^4} dt.$$ This converges for $\Re(z) > 0$, and the asymptotic expansion $$ ...
0
votes
1answer
17 views

Differential/Bessel integration show that question

Given $y_k=J_m(\sqrt{\lambda_k}x)$ and let $y(x,\lambda)=J_m(\sqrt{\lambda}x)$. I can't seem to compute this integration and show $\int^1_0({{\dfrac{d}{dx}(xy'_k)y-\dfrac{d}{dx}(xy')y_k}} ...
1
vote
3answers
120 views

Integral of $\frac{e^{-x^2}}{\sqrt{1-x^2}}$

I am stuck at an integral $$\int_0^{\frac{1}{3}}\frac{e^{-x^2}}{\sqrt{1-x^2}}dx$$ My attempt is substitute the $x=\sin t$, however there may be no primitive function of $e^{-\sin^2 t}$. So does this ...
10
votes
1answer
202 views

How to evaluate $I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$

Prima facie, this integral seems easy to calculate,but alas, this not's case $$I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$ The numerical value is I=-1.122690024730644497584272... How to ...
0
votes
0answers
17 views

Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$

Given two monotonically increasing continuous functions $f,g$ prove that $$(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx,\; b>a$$ what I have tried: Let $$h(x) = ...
5
votes
3answers
71 views

Is $\int_0^\infty x^{a-1} (1-x)^{b-1} e^{t-cx} dx$ integrable?

I am trying to evaluate the integral below. Is it even integrable? (Online integral solvers e.g. WolframAlpha could not solve the indefinite or the definite integral.) $$\int_0^\infty x^{a-1} ...
1
vote
5answers
71 views

Contour integration of cosine of a complex number

I am trying to find the value of $$ -\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} \cos\left(be^{i\theta}\right) \mathrm{d}\theta,$$ where $b$ is a real number. Any helps will be appreciated!
2
votes
2answers
41 views

Prove that these integrals are equal. How to complete the proof?

$$\int_0^xf(u)(x-u)^2du=2\int_0^x\left(\int_0^{u_2}\left(\int_0^{u_1}f(t)dt\right)du_1\right)du_2$$ Ok, I derived both parts 2 times wrt $x$ and got equal integrals. But I'm suspicious whether it is ...
0
votes
0answers
18 views

Sine and Bessel integral extension to imaginary argument

I found this integral in Gradshteyn-Ryzhik's book, $$ \int_a^\infty\ J_0\left(b\sqrt{x^2-a^2}\right)\ \sin(cx) \mathrm{d}x = \frac{\cos\left(a\sqrt{c^2-b^2}\right)}{\sqrt{c^2-b^2}}; ...
4
votes
6answers
113 views

Prove that there is a function $w > 0$ such that $\int_{0}^{1}w(x) dx \neq 0$ and $\frac{\int_{0}^{1}w(x)x^{2}dx}{\int_{0}^{1}w(x)dx} = \frac{1}{2}$?

I can see that $w(x) := x$ on $]0, \infty[$ suffices, but I am after a systematic analysis to see this, which I am incapable to do.
1
vote
1answer
37 views

$f_n(x)=n\sin^{2n+1}x\cos x$. Find $\lim_{n\to\infty}f_n(x)$

Let $f_n(x)=n\sin^{2n+1}x\cos x$. Then find the value of $\lim\limits_{n\to\infty}\displaystyle\int_0^{\pi/2}f_n(x)\;dx-\displaystyle\int_0^{\pi/2}(\lim\limits_{n\to\infty}f_n(x))\;dx$ My Thoughts: ...
2
votes
3answers
53 views

Integral of an inverse

Let $f(x)=x^3−2x^2+5$. Then find the integral $$\int_{37}^{149} \! f^{-1}(x) \, \mathrm{d}x$$ I know the inverse theorem for differentiation.( I don't think we can apply it here). Is there other ...
4
votes
2answers
236 views
+50

Any tips to solve this integral : $I_1 = \int \ln(x^2)e^{\sin(x)}\sin(x^{\cos(x)}) dx$

Background: I was making new expressions to see whether I could efficiently find their derivatives... After having done that, I've started trying to integrate most of them; obviously most of them ...
4
votes
1answer
44 views

Integral of sine multiplied by Bessel function with complicated argument

I need a help with integral below, $$ \int_0^\infty \sin(ax)\ J_0\left(b\sqrt{1+x^2}\right)\ \mathrm{d}x, $$ where $a,b > 0 $ and real, $J_0(x)$ is the zeroth-order of Bessel function of the first ...
0
votes
1answer
25 views

Find the area of the surface generated when the given curve is revolved about the x-axis

Find the area of the surface generated when the given curve is revolved about the x-axis The part of the curve $y=12x-2$ between the points $(\frac{5}{12},3)$ and $(\frac{13}{12},11)$ I understand ...
0
votes
1answer
15 views

Area of the surface when the curve is revolved about the x-axis

Find the area of the surface generated when the given curve is revolved about the x-axis. $y=2x+7$ on [0,4] This is what I have : $$S=\int_{a}^{b} 2π \left(f(x)\right)\sqrt{1+f'(x)^2} $$ ...
0
votes
0answers
35 views

Differential Equation result

So im doing a differential equation and I turned it into the integration form but now im having trouble at solving it, and cant find anything yet. The integral: $$\int \frac{ds}{\ln(s)+5}$$
0
votes
4answers
32 views

Integration with absolute values

Could someone help me solve this problem, I'm not sure where I'm going wrong $$ \int_{0}^{2\pi} |x-\pi| $$ We break the integral up since $[0,\pi] < 0$ and $[\pi-2\pi] > 0$. I then have $$ ...
0
votes
3answers
85 views

Solve this integral $ \frac{1}{2 \pi} \int_{0}^{2\pi} e^{\frac{i}{2} \left( n - m \right)x} \cos(x) dx$

I am supposed to calculate the integrals $$ \frac{1}{2 \pi} \int_{0}^{2\pi} e^{\frac{i}{2} \left( n - m \right)x} \cos(x) dx$$ and $$ \frac{1}{2 \pi} \int_{0}^{2\pi} e^{\frac{i}{2} \left( n - m ...