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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6
votes
3answers
145 views

Evaluating $ \int_{0}^{\infty}\frac{\ln (1+16x^2)}{1+25x^2}\mathrm d x$

how to solve such type of definite integration? I would like to see various methods to evaluate following integral $$ \int_{0}^{\infty}\frac{\ln (1+16x^2)}{1+25x^2}\mathrm d x$$
0
votes
1answer
47 views

If $f(0)=1$ and $f(x)=0$ for $x\ne0$ then $f$ is Riemann integrable on $[-1,1]$ with integral $0$

If $f(0)=1$ and $f(x)=0$ for $x\ne0$ then $f$ is Riemann integrable on $[-1,1]$ with integral $0$. I simply used the def of R integral and I think I proved that f is indeed R integrable. But I ...
0
votes
1answer
38 views

Sharpening a curve

I have a frequency domain graph as shown. I need to "sharpen" the curve to get a better response, and computing large butterworth orders is not possible on my machine. Hence, I would like to know if ...
1
vote
1answer
119 views

Hemisphere surmounted by cone

We have a rotationally symmetric solid, consisting of a half-ball of radius $R$, surmounted by a right circular cone of height $H$. If the centroid of the solid is in the half-ball part, it will stand ...
1
vote
1answer
61 views

Mean value theorem for integration

Can anyone hint or give the outline of the proof. I am a bit of confusing how to find $x_0$. Let $\phi(x) \geq 0$ for $x \in [a,b]$, and $\phi$ decreasing on $[a,b]$, let $h : [a,b] \rightarrow ...
2
votes
3answers
52 views

How do you integrate $\int_0^1 yn(1-y)^{n-1} \, dy$?

I'm trying to figure out how to evaluate this integral: $$\int_0^1 yn(1-y)^{n-1} \, dy = \frac{1}{n+1}$$ where n is an integer greater than 1. I just don't know where to begin. Can you help point ...
3
votes
3answers
42 views

Trouble understanding how $\int_c^d f \leq 0$ implies $f \leq 0$.

We are asked to suppose that we have a function $f:[a,b] \rightarrow \Bbb R$ which is continuous and has the property that $$\int_{c}^{d} f \leq 0 \quad \text{ whenever } a \leq c < d \leq b.$$ ...
3
votes
2answers
85 views

Evaluate $\int \ \frac{2}{{}x\sqrt{9x^2 - 25}} dx$

I'm trying to evaluate $$\int \ \frac{2}{{}x\sqrt{9x^2 - 25}} dx$$ So I know that if I had just $$\int \ \frac{2}{{}x\sqrt{9x^2 - 25}} dx$$ then I would be able to use a natural log rule ...
1
vote
1answer
46 views

Solution of integral equation

If $x$ is a real-valued, differentiable function of $t$, what is, and how do I find the solution of $$\int_a^b x(t) \frac{dx(t)}{dt} dt$$
0
votes
3answers
26 views

How would I go about integrating an improper integral with an absolute value in the denominator?

$\int^1_{-1} \frac{1}{\sqrt{\lvert{2x-x^2}\rvert}} dx$ I'm getting stumped. The integral is improper because at 0 the function does not exist. I am thinking of completing the square and doing a ...
2
votes
1answer
60 views

Find upper limit of normal distribution integration

Considering the normal distribution with standard deviation equals to 0.9 and mean 2.1: $$ P(X\leq a) = \frac{1}{0.9\sqrt{2\pi}}\int_{-\infty}^{a} e^{-\frac12\frac{(x-2.1)^2}{0.9^2}}\,dx $$ I must ...
4
votes
1answer
54 views

Odd Function Integral?

I know if $f$ is an odd function then $$\int_{-L}^L f(x)\:dx = 0$$ my question is, is the converse necessarily true? Intuitively, I feel it should be that by assuming that the integral with those ...
2
votes
1answer
89 views

Evaluating $\int _{-\pi}^{\pi}x^2cos(nx)dx $

Hello I'm trying to evaluate $$\int_{-\pi}^{\pi} x^2\cos(nx)dx$$ I understand you have to apply integration by parts twice but I always get zero and I know this is wrong. I always end up with ...
1
vote
1answer
42 views

measure of open set with measure Haar

By a Haar measure on a locall compact group (Hausdorff) we mean a positive measure $\mu$ (contains the borel set's) such that The measure $\mu$ is left invariant The measure μ is finite on every ...
0
votes
1answer
20 views

Inequality extension to the boundary of domain

Let $f$ be holomorphic on the unit disk $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$. Then I know that the function $|f|^3$ is subharmonic on $\mathbb{D}$. So for every $r<1$ I have by ...
1
vote
1answer
61 views

Incorrect indefinite integral on MATLAB?

any ideas why matlab is giving me an incorrect answer here? 1 set of commands syms x L N2 N2 = 6*x/L^2 - 2/L^2 N2 = (6*x)/L^2 - 2/L^2 expand(int(N2*N2)) ans = ...
5
votes
3answers
114 views

Closed form of $\int_{0}^{1}\frac{dx}{(x^2+a^2)\sqrt{x^2+b^2}}$

Is it possible to get a closed form of the following integral $$\Phi(a,b)=\int_{0}^{1}\frac{dx}{(x^2+a^2)\sqrt{x^2+b^2}}$$
-1
votes
1answer
72 views

$ \int\sqrt{tan(x)}\;dx$ [duplicate]

So, i'm trying...$ \int\sqrt{tan(x)}\;dx$ i'm trying solving for funtions inverse so, let $ u= \sqrt{tan(x)}$ and $ u^2=tan(x)$ , $x = arctan(u^2) $
2
votes
0answers
36 views

What are the X,Y coordinates of round beads strung along an Archimedian spiral of string?

I have a commercial application for which I have simplified the underlying mathematical problem to the following: There are 32 spherical beads on a string, each of diameter 'd'. Each bead is touching ...
1
vote
3answers
99 views

Evaluate $\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$ [duplicate]

Evaluate $$ \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$$ where $n\in\mathbb{N}$ This one is another intriguing question from my worksheet. I'm only allowed to use ...
0
votes
1answer
106 views

Recursive formula for integration by parts of given functions

I need to find, if it actually exists, a recursive formula for the following evaluations of indefinite integrals: \begin{align} I_{1,n}(x,R) &= \underset{n \,\text{terms}}{\underbrace{\int dx ...
0
votes
1answer
31 views

How do I find the limits of integration?

NOTE: the topic is Green's Theorem in the Plane I'm working on a problem that requires me to find the outward flux of the field: $$F = \left(3xy - \frac{x}{1+y^2}\right) \hat{i} + \left(e^x + ...
2
votes
2answers
103 views

Evaluate $\int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin x} \text{d}x$

Evaluate $$ \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin x} \text{d}x$$ where $n\in\mathbb{N}$ I was able to do this using parametization (differentiation under integration) but I'm not ...
1
vote
5answers
89 views

Compute $\int \frac{\mathrm{d}x}{49x^2+1}$

So I tried solving this by taking a substitute for the integrand, $t=49x$, so its derivative is $dx = \frac {dt} {49}$. Then you insert it into the integrand and get $$\int \frac{\mathrm dt}{49(t^2 ...
1
vote
3answers
42 views

How to find the integral of $\lim_ \ x^2 \arcsin(x^3)/\sqrt{1-x^6}\ dx$?

I could really use some help trying to find this integral please. $$\int \frac{x^2\arcsin(x^3)}{\sqrt{1-x^6}} dx$$
0
votes
1answer
35 views

Limit of Derivative and Derivative of Limit

Assuming the integral is finite and $f$ is continuous, does this argument always work? If not, what do we need more? $\displaystyle \frac{d}{dx}\int_x^{\infty} f(t) \, \mathrm{d}t =$ $\displaystyle ...
5
votes
1answer
266 views

Closed-form of $\int_0^{\pi/2}\frac{\sin^2x\arctan\left(\cos^2x\right)}{\sin^4x+\cos^4x}\,dx$

I have just seen two active posts about integrals of inverse trigonometric function, $\arctan(x)$, here on MSE. So I decide to post this question. This integral comes from a friend of mine (it's not a ...
0
votes
0answers
48 views

Integration by parts with Bessel function $j_0$

I need to prove this: $$ \mathcal F{\frac{1}{r^2}}\frac{d}{dr}r^2 \frac{dC}{dr}$$ $$= (\frac{2}{\pi})^{1/2} \int_0^\infty\frac{1}{r^2}\frac{d}{dr}r^2\frac{dC}{dr}j_0(kr)r^2dr$$ $$ =-k^2 ...
1
vote
3answers
33 views

series involving a logarithm of ${1\over ln^2(n)}$

$${\sum_{n=2}^\infty}= {1\over ln^2(n)}$$ Can I substitute ${x}$=${1\over ln(n)}$ and using the integral test, set it up to be $${\lim_{t\to infty}}= \int_2^tx^2 dx$$ and solve from there? and then ...
1
vote
2answers
65 views

Are all products of trigonometric functions integrable?

I have a feeling that this question has a really obvious answer, so forgive me if it turns out to be trivial. That being said, my question is whether all functions involving trigonometric functions ...
1
vote
0answers
28 views

Question about integration with a transcedental function

Lets say $\int$$x(e^x$$^2+2)dx$ my professor said that you need to distribute X first because its a transcedental function and not simply $u$$=x^2$ where $du=2x$. now is $lnx^2$ a transcedental ...
1
vote
3answers
130 views

Is integrating over complex numbers like this valid?

I had to evaluate the integral $$I=\int_{0}^{\pi} x^4 \sin{x} \ \mathrm{d}x$$ I thought that integrating by parts would be to long, and so, planning to use the property $\displaystyle\int e^{x} ...
0
votes
0answers
129 views

Related Rates problem about balloon and bicycle rates

Question: A balloon is rising vertically above a level , straight road at a constant rate of 1ft/s. Just when the balloon is 65 feet above the ground, a boy riding a bicycle is moving at a constant ...
0
votes
1answer
47 views

Evaluate $\displaystyle\int-x^{1-n}e^{xt}\ dx$

I have to evaluate $$\large\displaystyle\int-x^{1-n}e^{xt}\ dx$$ with respect to x but I am not sure how. I have tried integration by parts but this gets very complicated, is there an easier way? ...
0
votes
1answer
45 views

Compute three-dimensional integral

${\int\int\int}zdxdydz$ over $M$, where $M = [x,y], x \ge 0, y\ge 0, z\ge \sqrt{x^2 + y^2}, x^2 + y^2 + z^2 \le 2x.$I guess that $0 \le x \le 2, 0 \le z$, but I have no idea how to guess $y \le f(x)$ ...
2
votes
4answers
59 views

Prove that $\int_{-1}^{1} 4x \sqrt{1-x^{2}}\, dx = -3\pi$

What is of interest is the assertion $$\int_{-1}^{1} 4x \sqrt{1-x^{2}}\,dx = -3\pi.$$ Since $$\pi = 2\int_{-1}^{1}\sqrt{1-x^{2}}\,dx,$$ i.e. the area of a unit circle, it suffices to prove that ...
7
votes
4answers
394 views

Finding $\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$

$$\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$$ I try to solve it, but failed. Who can help me to find it? I encountered this integral when trying to solve ...
2
votes
2answers
56 views

Evaluate $\int\frac{dx}{x(2-\ln x^2)}$

How to evaluate $\displaystyle\int\frac{dx}{x(2-\ln x^2)}$? I tried to use an operator, but before that I distributed so $\displaystyle\int\frac{dx}{(2x-x\ln x^2)}$ my operator was $-x\ln x^2$ am I ...
0
votes
1answer
66 views

Find the Fourier transform of $\frac1{1+t^2}$

Find the Fourier transform of $$f(t)=\frac1{1+t^2}$$ using contour integration that $$F\{f(t)\}=\int^\infty_{-\infty}\frac1{1+t^2}e^{2\pi ft}dt$$ How can I do this?
1
vote
0answers
30 views

how to solve the complex Integration

how to solve the integral $$\int_{-\lambda_1\cdot x_1}^{-\lambda_1^2}\frac{e^z(1-e^z)^{n-1})}z dz$$ I have to find entropy of paramter of the exponential distribution under different loss functions. ...
1
vote
1answer
38 views

What is the substitution to remove integration limit from inside an integral?

For instance, how to remove the $m$ from inside this integral? $$\int_0^m f\left(\frac rm\right) dr$$
0
votes
2answers
59 views

Prove that $\lim_{m\to\infty} I_{2m}/I_{2m+1}=1$

$I_n$ is given by $\int ^ {\pi/2}_{0} \sin^n(x)\, dx$ Prove $I_{2m+1} \leq I_{2m} \leq (1+ \frac {1}{2m}) I_{2m+1}$. Also prove $\displaystyle \lim_{m\to\infty} \frac {I_{2m}}{I_{2m+1}}=1$. I'm ...
1
vote
2answers
76 views

Integral of exponential

I have been reviewing complex analysis since it's been awhile since I dove into it I am stumped of the following integral, $$\int^{\infty}_{-\infty}e^{iax-bx^2}dx.$$ By setting $z=x-i\frac{a}{2b}$, I ...
6
votes
2answers
77 views

How to solve $\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$

Could you help me to prove $$\frac{\partial{\rm B}}{\partial b}\left(0^+,1\right)=-\frac{\pi^2}{6}$$ where ${\rm B}(a,b)$ is Beta function.
1
vote
1answer
48 views

Prove $(n + 1)I(n) = (n + 2)I(n + 2)$ by integration by parts.

$I_n$ is given by $\int ^ {\pi/2}_{0} \sin^n(x) dx$ My attempt: I got to the fact that the statement is true iff $$\dfrac {n+1}{n+2} = \int ^ {\pi/2}_{0} \dfrac {\sin^2(x)}{\sin^n(x)}$$ I do not ...
1
vote
2answers
112 views

Evaluating $\int\frac{4x^3-3x^2+6x-27}{x^4+9x^2}dx$

$$\int\frac{4x^3-3x^2+6x-27}{x^4+9x^2}dx$$ this integral get very messy. Can I get a step by step breakdown of solving?
-4
votes
1answer
41 views

Messy Integral of polynomial over polynomial

$\int (x^3+6x^2+3x+16)/(x^3+4x) dx $ This integral gets very messy. Can I get a step by step break down of how to solve?
3
votes
2answers
69 views

Integral of constant divided by polynomial and another constant

$$\int_{-\infty}^{-1}\frac{4}{\sqrt{x^6+2}}\,dx$$ What are the steps to integrate?
3
votes
1answer
82 views

If $U(f,P) = L(f,P)$, show that $f$ is constant.

The question has two part, Show that if f : [a, b] → R is continuous and there exists a partition P of [a,b] such that U(f,P) = L(f,P), then f is constant. Is this true if we drop the assumption ...
1
vote
1answer
66 views

Definite integral reciprocal of polynomial plus trig function

Evaluate$$\int_0^{\pi/2}\frac{dx}{x^{1/4}+\cos x}$$ What are the steps to integrate?