Tagged Questions

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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2
votes
1answer
18 views

using contour integrals

Let $ \gamma (t)= e^{it} $ where $0 \leq t \leq 2 \pi.$ Evaluate $\int_{\gamma}$ $e^{z}$ $dz$ . Use the result to show that $\int_{0}^{2\pi} e^{\cos(t)}\cos(t+ \sin(t)) dt = 0$. I have worked out ...
2
votes
0answers
19 views

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
3
votes
2answers
69 views

A hard Integration question

How would I integrate this; $$\int-e^{\cos(t)}\sin(\sin(t)+t)dt $$ I have tried several methods but cant seem to work this out.
0
votes
0answers
18 views

How to find The velocity of a sprinter on a straight track

How to find The velocity v(t) of a sprinter on a straight track is shown in the graph below. ? I know how to take anti der, but how do I get the numbers? Please help. Thanks ...
0
votes
1answer
14 views

Evaluating a double intgeral over a plane region

I'm trying to evaluate the following double integral:$$\iint_{D}\frac{1}{\sqrt{4-x^2-y^2}}\,dA$$where $D$ is the disk of radius 1 with center at $(0,1)$. Is there a useful substitution to make? If I ...
-2
votes
3answers
38 views

Evaluating $ \int\frac{x}{\sqrt{3-x^2-2kx}}\,dx $ [on hold]

I'm trying to evaluate this integral: $$ \int\frac{x}{\sqrt{3-x^2-2kx}}\,dx $$ where $k$ is a real parameter.
2
votes
0answers
56 views

Is this limit finite?

What is the limit of $$\lim_{u+v\rightarrow 1}\frac{\ln \int f_0(y)^{1-v} f_1(y)^{v}\mathrm{d}y- \ln \int f_0(y)^u f_1(y)^{1-u}\mathrm{d}y}{1-(u+v)}$$ where $f_0$ and $f_1$ are some density ...
3
votes
3answers
45 views

Apply dominated convergence theorem to show differentiability

Let $f,g \in L^p(\mu), 1 < p < \infty$. Show that the function $$\phi(t)=\int |f+tg|^pd\mu$$ be differentiable at $t=0$ and find $\phi'(0)$. My try, $\psi(t)=|f+tg|^p$ is differentiable at ...
1
vote
5answers
71 views

Indefinite integral of $\frac{\sqrt{x}}{\sqrt{x}+1}$

For this I tried using the substitution technique, but it got me nowhere near the right answer. What my notepad looks like: $$f(x) = \dfrac{\sqrt{x}}{\sqrt{x}+1}$$ and $$F(x) = \int f(x) = ...
2
votes
3answers
61 views

Integral of $e^{\frac{y}{x}}$

How can we find $\int e^{\frac{y}{x}}dy$. An explanation of the answer would be helpful. The answer I got is $ x e^{y/x}$. But not sure about the steps used for obtaining the answer...
3
votes
2answers
35 views

evaluation of this logarithmic integrals

what is the value $$ \int_{a}^{\infty} \frac{\log^{n}(x)}{x^{2}}\mathrm{d}x $$ 'a' is a positive integer and so is 'n' my gues with a change of variable $ x=e^{t} $ is that this integral would be ...
3
votes
1answer
33 views

A function defined by $L^p$ integral is continuous on the boundary

Suppose $f$ is a measurable function on $X$, $\mu$ is a positive measure on $X$, and $$g(p)=\int_X|f|^p d\mu=||f||_p^p, (0<p<\infty)$$ Let $E=\{p|g(p)<\infty\}$. Assume $||f||_\infty >0$. ...
5
votes
2answers
40 views

Find $\mathcal{L}\left\{\cos^3\left(t\right)\right\}$

I began by breaking the problem up as follows: \begin{align} \mathcal{L}\left\{\cos^3\left(t\right)\right\}=\int_0^\infty e^{-st}\cos^3\left(t\right)\:dt & = \int_0^\infty ...
2
votes
0answers
13 views

Integration with respect to signed measure and Radon Nikodym theorem

I aim to show the following question: Let $\mu$ be a $\sigma$-finite measure, and $\lambda$ a finite signed measure on $(X,M)$ satisfying $\lambda\ll\mu$, let $h=\frac{d\lambda}{d\mu}\in ...
0
votes
3answers
39 views

Proof with Fundamental Theorem of Calculus

If $f'(t)\leq 10$ for $0 \leq t \leq 5$ and $f(0)=3$: How I can explain with $\int_a^b f'(t)\,dt=f(b)-f(a)$ what the maximum of $f(5)$ is?
0
votes
2answers
29 views

Proof with integral properties

I'm trying to explain/looking for an answer whether a positive function $u=f(t)$ exists, for which $\int_{t=0}^{1}u\,dt = \int_{t=1}^{0}u\,dt$ is true. As we all know, the correct theorem is ...
7
votes
1answer
185 views

Show the sequence converges to M

Assume $f : [a,b] \to R$ is continuous and $f(x) \ge 0$ for all $x \in [a,b]$, and $M = \sup\{f(x) : x \in [a,b]\}$. Show that $$\lim_{n\to\infty}\left[\int_a^bf(x)^ndx\right]^{1/n}$$ converges to ...
-2
votes
2answers
49 views

Find $\int_0^{2\sqrt{\pi}}\int_{x/2}^{\sqrt{\pi}}\sin(y^2)dydx$

Find $$\int_0^{2\sqrt{\pi}}\int_{x/2}^{\sqrt{\pi}}\sin(y^2)dydx$$ Not sure how to start
1
vote
1answer
28 views

Another Fundamental Theorem of Calculus Proof

Let $f : R \to R$ be continuous and $\delta > 0$. Define $g(t)=\int_{t-\delta}^{t+\delta}f(x)dx$ for all $t \in R$. Prove that $g$ is differentiable and compute $g'$. I'm pretty sure you know that ...
1
vote
2answers
328 views

Fundamental Theorem of Calculus Proof

Find $f'$ where is $f$ is defined on $[0, 1]$ as indicated: $$f(x) = \int_x^{\sqrt{x}} \frac 1{1+t^3}dt$$ I know that the fundamental theorem is going to be used in this proof, but I'm not really sure ...
1
vote
2answers
28 views

Derivative of a definite integral with two constraints

I am new to this, and I want to see if I have the right answer: $$\frac{d}{dx}\int^{2x}_{x} s^2 ds = \int^{0}_{x} + \int^{2x}_{0}= -\int^{x}_{0}+\int^{2x}_{0} =-x^2+8x^3 $$
1
vote
2answers
38 views

Estimating integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$

Estimate the definite integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$ with an error of at most $10^{-4}$, using the Alternating Series Estimation Theorem. My approach is as follows: I found the ...
2
votes
3answers
39 views

Derivative of a definite integral with fraction

I think I may be on the right track but need some help pulling my thoughts together. I have this problem: $$\frac{d}{dt}\int^{\frac{1}{t}}_0 \frac{dx}{1+x^2}$$ So, I believe I want to spread this ...
14
votes
6answers
187 views

Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques

Is there a way to compute this integral, $$\int_0^1 \frac{x^k-1}{\ln x}dx =\ln({k+1})$$ without using the derivation under the integral sign nor transforming it to a double integral and then ...
0
votes
0answers
25 views

Show f is integrable and integral is C(b-a)

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$ Consider for real $a<b$ and real $A,B,C$, the function ...
1
vote
0answers
45 views

Problem in understanding the process of finding antiderivative.

Antiderivative or indefinite integral is the family of functions the derivative of which gives the original function. Now, let's elaborate the process. Suppose $F(x)$ is the derivative of the ...
0
votes
0answers
19 views

How to evaluate this line integral?

I'm given: the integral of (x+y) ds, it's a straight line segment x=t, y=(1-t), and z=0. From (0,1,0) to (1,0,0). Help is appreciated!
0
votes
1answer
27 views

Why are integral and differential operators commutative?

For instance, let's assume a constant 3D surface over time $S$. $$ \frac{d}{dt}\iint_S \mathbf B \cdot \mathbf{ds} \quad=\quad \iint_S\frac{\partial \mathbf B}{\partial t}\cdot \mathbf{ds} $$ Why ...
4
votes
1answer
34 views

One Question about the Fubini's Theorem

The Fubini's Theorem says: If function $f:X \times Y \rightarrow R$ is integrable over $X \times Y$, then $$ \int_{X \times Y}f(x,y)dxdy = \int_{X}dx\int_{Y}f(x,y)dy = \int_{Y}dy\int_{X}f(x,y)dx. $$ ...
0
votes
0answers
18 views

Dirac delta distribution and vector integral.

I am trying find $\int_{all space} d^{3}q \delta^{3}(\vec{q})\frac{(\vec{p}\cdot\vec{q})^2}{q^{2}}$ where $\vec{p}$ is some fixed vector. Is there any way to find this integral? Thanks
-2
votes
2answers
58 views

Consider the integral.

So I missed class today and decided to take a look at the homework assigned. This notation is unfamiliar to me. Up until this point, we have just been finding over and underestimates based on ...
0
votes
0answers
16 views

Non-linear integral equation

Show that the function $$x(t) = \frac{1}{{\sqrt {k \cdot m} }} \cdot \int_0^t {F(\tau ) \cdot \sin \left( {\sqrt {\frac{k}{m}} \cdot (t - \tau )} \right)\,d\tau } $$ satisfies the initial conditions ...
0
votes
1answer
13 views

Evaluate the integral (using partial fractions maybe?) [duplicate]

Evaluate the following integral $\int{\frac{1}{(x+a)(x+b)}}$ (this might involve partial fraction decomposition, $\int{\frac{1}{x^2+x(a+b)+ab}}$ this is what my first step was)
0
votes
1answer
29 views

Integration of logarithms

Integral of $$1\over x(log_4^x)^2$$. I changed $$(log_4^x)^2$$ to $$2(ln(4)/(ln(x))$$ Then I integrated and got $$.5ln(4)(ln(ln(x))$$ The answer to the problem is $$ -ln(4)/(log_4^x) + C...$$ ...
6
votes
3answers
98 views

Evaluate $\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx$ (solution verification)

I tried to find the integral $$I=\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx \tag1$$ I substituted $x=t^2, 2tdt=dx$ and chose $\log x$ and $\sqrt{x}$ to be principal values. We have ...
1
vote
1answer
19 views

Expectation of the derivative of a random process

Let's have a Random Process Y(t) = X(t) + 0.3 X'(t) Mean of X(t) = 5t Question : Find the mean function of Y(t) What I did : E(Y) = E(X) + 0.3E(X') ? I don't know if i have to take the derivative ...
1
vote
1answer
40 views

How to show that continuous functions in $L^1$ is Riemann integrable

How to show that for continuous functions in $L^1$ is Riemann integrable in other words. \begin{align*} \int_{\mathbb{R}} f \, d \lambda=\int_{-\infty}^\infty f \, dx \end{align*} I already showed ...
1
vote
0answers
10 views

Functional derivative of a repeated integral

For a given function $f$, the functional derivative of the functional $\mathcal{F}[\rho]=\int f(x,\rho(x))\,dx$ is well-known to be $\frac{\delta}{\delta \rho(x)}\mathcal{F}[\rho]=\frac{\partial ...
3
votes
2answers
67 views

Closed form for $\int_0^\infty e^{-x}\sin^a(x)dx$

Can we find a closed form for $$I(a)=\int_0^\infty e^{-x}\sin^a(x)dx$$ Mathematica can easily find closed form for integer $a$: \begin{align*} I(0)&=1\\ I(1)&=1/2\\ I(2)&=2/5\\ ...
0
votes
2answers
30 views

Showing integrability

Define the function $f\colon [0,1] \to \mathbb{R}$ by $$f(x) = \begin{cases} 1 & x = \frac{1}{n}, \, n \in \mathbb{Z}\\ 0 & \text{otherwise} \end{cases} $$ The ...
0
votes
2answers
19 views

Translating expected values between two sets of related iid variables

The setting: $\mu$ is a probability measure on $\mathbb{R}$, $f: \mathbb{R} \to [0, \infty)$ so that $0 < ||f||_{L^1(\mu)} < \infty$, and $v$ is another probability measure defined by $v(A) = ...
3
votes
1answer
42 views

What am I doing with this triple integral?

I am new here and hope my question is clear and is straight to the point. The following is a form of an integral I am trying to compute. $$\int_{x}\int_{y}\int_{z} f(x,y,z) g(x,y)\ dz \ dy \ dx \ ...
3
votes
5answers
56 views

Find the equation of a curve where $\frac{dy}{dx}=2x+y$ at all points

Find the equation of the curve which passes through the origin and is such that $\frac{dy}{dx}=2x+y$ at all points $(x,y)$ on the curve, giving the equation in the form $y=f(x)$. I checked with ...
0
votes
1answer
20 views

How to show that $\lim_{h\to 0}\int_0^h|f(x)|dx=0.$

Let $f: \mathbb{R} \to \mathbb{R}$ be a locally integrable function. How can we see that $$\lim_{h\to 0}\int_0^h|f(x)|dx=0.$$ If $f$ is bounded, then we have the result. But what about $f$ only ...
4
votes
1answer
63 views

Evaluating $\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$ using complex analysis

how do I compute $$\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$$ with complex analysis? I feel like im calculating the residue wrong and I cant get to the answer correctly. I tried to branch cut ...
0
votes
1answer
17 views

Integral with moving average

Hi everyone, My buddies and I are stuck on this question. What is this question even asking for? We know it has to do something with taking the integral over different sections. But how do we go ...
0
votes
0answers
23 views

circular contour integral with complex numbers [on hold]

Let gamma(w,R) denote the circular contour t maps to w + Re^it where 0 < t < 2Pi. Evaluate the integral of 1/1+z^2 when gamma is gamma(i; 1)
0
votes
1answer
33 views

Example of function that is not $L^1$ such that Riemann and Lebesgue integrals are not equal

What would be an example of function that is not $L^1$ such that Riemann and Lebesgue integrals are not equal? Thanks
0
votes
2answers
35 views

Integration ( Area of a shaded region )

I got stuck on this question for quite some time but I still can't get my head around it especially for question b). Please help. The curve $C$, shown in Figure 2, has equation $y = 3x – x^2$ It ...
-3
votes
2answers
86 views

Evaluating $\int 2x e^{x^2} \, dx$ [on hold]

Can I have some pointers on how I should evaluate following indefinite integral? $$\int 2x e^{x^2} \, dx$$