All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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1answer
118 views

Series around $s=1$ for an integral

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
3
votes
1answer
84 views

Integrating a differential equation?

How does $(xJ_0'(x))'+xJ_0(x)=0\implies\int_0^1 x J_0(ax)J_0(bx) dx={bJ_0(a)J_0'(b)-aJ_0(b)J_0'(a)\over{a^2-b^2}}?$ Thanks. Perhaps int by parts? But how do I get the RHS form?
3
votes
2answers
411 views

Simple integration by parts problem?

An integration reduction formula for $\sec^n x$ is $$\int\sec^nx\;dx=\frac{1}{n-1}\sec^{n-2}x\tan x+\frac{n-2}{n-1}\int\sec^{n-2}x\;dx, n≠1$$ Using this formula (which I am sure is correct) gives ...
3
votes
3answers
5k views

Tail sum for expectation

In Pitman's Probability, the tail sum formula for expectation is introduced for a nonnegative (0,1,...) discrete random variable $X$: $$E(X) = \sum_{i=0}^\infty P(X > i).$$ I wonder if there is ...
3
votes
1answer
373 views

On the absolute integrability of Bessel functions

Reading "How do you integrate a Bessel function", it didn't seem like it was an easy task. Thinking more about Bessel functions, speficially $J_0(x)$, it occurred that it looked a lot like the sinc ...
3
votes
2answers
100 views

Quick inverse trigonometric integration question.

I am a little confused here, how does removing $1/2$ from the function to the outside of the integral get rid of the $t$ in the numerator in this problem? $$\eqalign{ \int\dfrac t{t^4+25}dt & = ...
3
votes
2answers
867 views

Compute unknown limit from known integral in Mathematica?

I have an integral $\int_a^b \! f(x) \, \mathrm{d}x = c$ where I know c and either a or b. Now I want to compute either b or a (i.e., the missing limit). How would I do that in Mathematica/Wolfram ...
3
votes
3answers
254 views

Multiple integral Issue

I'm given the following exercise: $\iint\limits_D \exp(x^{2}+y^{2})dA$ And I dont even know where to start, any chance someone could give me a hint? D is a half circle, given by: $9\le ...
3
votes
3answers
65 views

What is the value of $ \int_{x}^{1} \arcsin \left( \frac{2t}{t^2+1} \right) \text{d}t $?

Is this result true? Wolfram doesn't seem to be able to evaluate the definite integral in the allowed time. $$ \int_{x}^{1} \arcsin \left( \dfrac{2t}{t^2+1} \right) \text{d}t = \dfrac{\pi}{2} - ...
3
votes
3answers
98 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
3
votes
2answers
47 views

Comparison of Newton-Cotes Quadrature and Gaussian Quadrature formulas

Newton-Cotes quadrature formulas are a generalization of trapezoidal and Simpson's rule. The trapezoidal rule involves $2$ points, Simpson's rule involves $3$, and in general Newton-Cotes formulas ...
3
votes
2answers
91 views

What does the Fourier transform of $1/x^2$ mean?

If I ask Mathematica to compute the Fourier transform of $\frac{1}{x^2}$ using the FourierTransform function, it gives me a result of ...
3
votes
2answers
55 views

Infinitely real-differentiable function with $f(0)=0$ but $\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$

I'm searching for a infinitely real-differentiable function $f:\mathbb{C}\to\mathbb{C}$ with $f(0)=0$ but $$(*)\;\;\;\;\;\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$$ where ...
3
votes
1answer
108 views

What is the meaning of $dA$ in double integrals?

What is the meaning of $dA$ in $\iint_E\dots dA$, where $E$ is a region in the $xy$ plane? In some integrals we use $dA=dx\,dy$, but in others $dA=\hat{k}\,dx\,dy$. (Here $\hat {k}$ is the unit ...
3
votes
1answer
123 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 ...
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votes
2answers
67 views

Calculation of integral $\int\exp \left(-\alpha \sin^2 \left(\frac{x}{2} \right) \right) dx$

Given $\alpha$ is a constant. How to calculate the following integral? \begin{equation} \int \exp \bigg(-\alpha \sin^2 \bigg(\frac{x}{2} \bigg) \bigg) dx \end{equation} Thanks for your answer.
3
votes
2answers
102 views

Evaluate the definite integral $ \int_{-\infty}^{\infty} \frac{\cos(x)}{x^4 +1} \ \ dx $

I am having more trouble with this problem then I feel like I should be. I set $ \ \cos(x) = e^{ix} \ $ and $ \ x^4 +1 = e^{\pi i /4} \ $ or $ \sqrt{i} \ $. I think I am suppose to do a residue to ...
3
votes
1answer
120 views

Need help evaluating a definite integral $\displaystyle \int^{\infty}_{-\infty} \frac{dx}{(1+4 x^2)\cosh(\pi x)} = \ln2$

Can anyone show how to evaluate this integral? $\displaystyle \int^{\infty}_{-\infty} \frac{dx}{(1+4 x^2)\cosh(\pi x)} = \ln2$
3
votes
2answers
64 views

How to prove divergence of the integral $\int_{0}^{\infty}\frac{\sin(x)}{x^{2}}dx$

I want to show that the following integral diverges: $$\int_{0}^{\infty}\frac{\sin(x)}{x^{2}}dx$$ I used the substitution $ t = \frac{1}{x} $ to transform this integral into $$\int_{0}^{\infty}\sin ...
3
votes
2answers
80 views

How to solve the integral $\int_0^1\frac{\cos x \ln x}{\sqrt{x}}$?

I am trying to solve the following integral: $\int_0^1\frac{\cos x \ln x}{\sqrt{x}}$ On wolfram-alpha I get the approximated value: -3.92203 Can anyone help me? Thanks in advance!
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votes
2answers
40 views

An integral I got on my midterm exam

Derive a recursive formula for the integral $I(n) = \int_0^1x^{m}\ln^{n}(x)\,dx$ and then solve the integral for $m = 0$. I have tried using partial integration as follows: $$ u = \ln^{n}(x) ...
3
votes
1answer
90 views

Fundamental Theorem of Calculus when Integrand is a Function of the Bounds

I have a function, $$F(r) = \int_0^r |c x^2 + {(2 a + b - 4 a r - 3 b r - 2 c r) x^2\over2 r} + b x^3 + a x^4| dx$$ a, b and c are constants. I want to determine r such that $f=F'(r) = k$. ...
3
votes
3answers
50 views

How find this equation $y''+(y')^2\cdot e^x=0$

if the ODE $$y''+(y')^2\cdot e^x=0$$ such $$y(0)=1,y'(0)=1$$ Find the $y(x)=?$ my ugly methods: let $$y'=p,y''(x)=p'(x)$$ so $$p'(x)+p^2\cdot e^x=0$$ $$\dfrac{dp}{dx}=-p^2e^x$$ ...
3
votes
2answers
50 views

How Do I Integrate? $\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}$

How do I integrate this one? $$\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}\,dx$$ Is my answer correct: $$-3\ln\left \| x+2 \right \|+\ln\left \| x \right \|+\frac{4}{x}+C$$
3
votes
2answers
160 views

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
3
votes
3answers
87 views

Evaluate $\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = \sec(u)$ substitution: $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &=& \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &=& ...
3
votes
2answers
93 views

Why this equation is true?

Pardon my ignorance. I don't know enough calculus to understand this. I assume this is a very easy question for this amazing site. I saw this on the The Theory of Riemann Zeta Function Book. ...
3
votes
1answer
64 views

Integral inequality problem

Let $f:[0,1]\to\mathbb R$ be a differentiable function with $f(0)=0$ and $f'(x)\in(0,1)$ for every $x\in(0,1).$ Show that $$\left(\int_0^1f(x)dx\right)^2>\int_0^1(f(x))^3dx$$ I am not even sure ...
3
votes
2answers
134 views

Integral $\int_0^\infty e^{imx^2}dx$

In evaluating an integral in path integrals in QFT, I am stuck with this integral (that came up from evaluating a functional integral), $$I = \bigg( \frac{m}{2\pi i\tau}\bigg) \int ...
3
votes
2answers
117 views

How to find $\int_2^x t/(\log t)^2 \,dt$

$$\int_2^x \frac{t}{(\log t)^2} \,dt,$$ I want to write this integral with $Li(x)$ or $Li_2(x)$. How can i do that?
3
votes
1answer
54 views

Help with integral $\int^{a\sqrt{1-E/V}}_{-a\sqrt{1-E/V}}\sqrt{2mV(1-x^2/a^2)-E} dx $

Hi I have tried u and trig substitution for this integral and just cant get it can someone offer a pointer or two? Thanks $\int^{a\sqrt{1-E/V_0}}_{-a\sqrt{1-E/V_0}}\sqrt{2m[V_0(1-x^2/a^2)-E]} dx $ ...
3
votes
1answer
49 views

Improper integral problem. $\int_9^\infty dx / [(x-8) (x-7)]$

I worked the problem out and ended up with $\ln(x-8) - \ln (x-7)$. So according to the natural log rule, I can make the problem $\ln (x-8)/(x-7)$. And then when I plug 9 in, I end up with 1/2 as the ...
3
votes
2answers
55 views

Integrate the following function:

Evaluate: $$\int \frac{1}{ \cos^4x+ \sin^4x}dx$$ Tried making numerator $\sin^2x+\cos^2x$ making numerator $(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x$ Dividing throughout by $cos^4x$ Thank you in ...
3
votes
3answers
103 views

Show $f$ in Riemann integrable

Let $\displaystyle f(x)=\begin{cases} \frac{1}{n}, & \text{if }x=\frac{m}{n},m,n\in\mathbb{N}\text{ and m and n has no common divisor} \\ 0, & \text{otherwise} \end{cases}$ Show $f\in ...
3
votes
3answers
63 views

Question about Integration by Parts with e^x and sin function

I have this Integration by Parts question that I can't seem to find an answer to. The question is: $$\int\ e^x \sin(9x)\,dx$$ I used u-substitution: $$u=e^x,du=e^x\,dx$$ $$dv=\sin(9x)\,dx, ...
3
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1answer
146 views

A calculus counterexample!

Give me an example of two Riemann-integrable functions $f,g:[0,1]\to[0,1]$ such that $g\circ f$ isn't integrable! I already know the following example: $$f(x)=\begin{cases} 0, & \text{if ...
3
votes
1answer
92 views

Proving Legendres Relation for elliptic curves

The legendre's relation can be stated as follows $$ K(k) E(k^*)+ E(k) K(k^*) - K(k) K(k^*) = \frac{\pi}{2} $$ where $k^* = \sqrt{1 - k^2}$ is the complimentary modulus, and $E$ and $K$ are ...
3
votes
1answer
35 views

Sine substitution?

My book says the following: $$\int \frac{dx}{(16-x^2)^{3/2}}$$ $$x = 4\sin\theta$$ $$(16 - x^2)^{3/2} = (4^2\cos^2\theta)^{3/2}$$ $$=(4\cos\theta)^3$$ I don't understand the last step: Doesn't: ...
3
votes
3answers
60 views

Prove that $\frac{d^n}{dx^n}\left(\frac{\sin x}{x}\right)=\frac{1}{x^{n+1}}\int_0^x y^n\cos\left(y+\frac{n\pi}{2}\right) \, dy,\: n\in \mathbb{N}$

Prove that $$\frac{d^n}{dx^n}\left(\frac{\sin x}{x}\right)=\frac{1}{x^{n+1}}\int_0^x y^n\cos\left(y+\frac{n\pi}{2}\right) \, dy,\: n\in \mathbb{N}$$ My try $n=0$ then $\frac{\sin x}{x}=\frac{1}{x} ...
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votes
3answers
1k views

Integral of cos(5x)cos(3x)cos(4x)dx

The integral $\int_0^{\pi/8}\cos(5x)\cos(3x)\cos(4x) \,dx$ is equal to $k/24$. Find the constant $k$. So far, I assume that the best way to solve this question is to solve the integral and compare ...
3
votes
1answer
96 views

Integration by parts seems to be infinite, but answer is simple

I have to integrate this: $$ \int_{-1}^0 \int_{-2x-2}^{2x+2} x^2 y^2 + \sin(xy) e^{x^2} y^2 \;\operatorname d\!y \operatorname d\!x $$ I started integrating by parts, $$ \int_{-1}^0 e^{x^2} ...
3
votes
2answers
49 views

Proving an integral using a series

If $f:(0,1]\rightarrow \textbf {R}$ is defined by $f(x)=2nx$ for $\frac{1}{n+1}\leq x \leq \frac 1n$ and $n$ is a natural number, assuming that $\sum_{k=1}^{k=\infty}1/k^2=\pi^2/6$, show that ...
3
votes
2answers
111 views

Improper parametric integral and differentiation under the integral sign

While looking at an astrophysic problem, I encountered the following integral $$ \rho_{\infty} (r) = \int_{r}^{a} \frac{\rho_{0} (r_{0})}{\sqrt{r_{0}^{2} - r^{2}}} d r_{0} \;\;\;\;\;\;\; (1)$$ The ...
3
votes
4answers
127 views

How come does integral of $\ln^2 x$ from $0$ to $4$ , converges?

Would someone please help my understanding how come integral of $\ln^2 x$ from $0$ to $4$, converges? Thank you
3
votes
3answers
72 views

Integration Techniques - Adding [arbitrary] values to the numerator.

Suppose you wanted to evaluate the following integral. Where did the 4 come from? I understand that it makes the solution but how would you make an educated guess to put a 4? And how in the future ...
3
votes
2answers
161 views

Indefinite Integral Questions.

Evaluate this indefinite integral. $I= \int{x\sqrt{4x+1}}dx$ let $u=4x+1$ $\frac{du}{dx}=4\rightarrow{dx=\frac{du}{4}}$ $I=\int{x}\sqrt{u}\frac{1}{4}du=\frac{1}{4}\int{x}\sqrt{u}du$ Then I ...
3
votes
2answers
104 views

Problem with integral.

How can I evaluate this integral? $$ \int{x^{3}\,{\rm d}x \over \left(x - 1\right)^{2}\sqrt{x^{2} + 2x + 4}}$$ I would be grateful for any tips.
3
votes
1answer
93 views

how to double integrate this exponential function

how can i integrate this $\int_0^∞\int_x^∞{\frac{e^{-y}}{y}}dy dx$ i am stuck here $\int{\frac{e^{-y}}{y}}dy$. I have tried some log substitutions.
3
votes
2answers
470 views

If $f$ and $g$ are Riemann integrable, are $f\cdot g$ and $f/g$ Riemann integrable?

I do not think they are, but I cannot seem to come up with a definitive answer. I have tried using the "Cauchy criterion" for integrability $$U(f,P)-L(f,P)<\varepsilon$$ Here, $U(f,P)$ is the upper ...
3
votes
2answers
76 views

Try to solve the following differential equation: $2y''=e^y$

I am trying to solve this equation: $$2y''=e^y$$ No $x$ in equation so: $$y''=P'P , y'=p \\ \implies 2P'P=e^y$$ After the integrating on both sides I got: $$P^2=e^y$$ and back to $y$: $$y'2=e^y \\ ...