0
votes
4answers
31 views

How can I show that these integrals are zero

How can I show that these integrals equal $0$ when $n$ and $m$ are both integers and $n \neq m$? $$\int_{-\pi}^{\pi}\sin(mx)\sin(nx)dx = \int_{-\pi}^{\pi}\cos(mx)\cos(nx)dx = 0$$ I'm able to show that ...
1
vote
1answer
22 views

Any Easier way to integrate:$\iint\limits_D{e^{x+y}}d\sigma,D=\{\left . (x,y) \right ||x|+|y|\leqslant1\}$

This is my way: \begin{align} \iint\limits_D{e^{x+y}}d\sigma & = \int_{-1}^0e^xdx\int_{-x-1}^{x+1}e^ydy + \int_0^1e^xdx\int_{x-1}^{-x+1}e^ydy \\ & = \cdots \\ & = e-e^{-1} ...
1
vote
0answers
29 views

How do I do this double integral (change of variable)

$B$ is the region bounded by $xy = 1$, $xy = 3$, $x^2 - y^2 = 1$, $x^2 - y^2 = 4$ Find $$\iint\limits_{B}x^2 + y^2 \,dx\,dy$$ using the change of variables: $$u = x^2 - y^2$$ $$v = xy$$ So I think ...
0
votes
1answer
31 views

Proving integration formulas from scratch

Prove the following integration formulas from scratch? (I uploaded them)
0
votes
1answer
35 views

How to calculate “general” integral $\int\limits_{a}^{b}f(x)^2dx$?

How to calculate "general" integral: $\int\limits_{a}^{b}f(x)^2dx$?
5
votes
4answers
79 views

Integral: $\int_0^{\pi/12} \ln(\tan x)\,dx$

I am trying to evaluate: $$\int_0^{\pi/12} \ln(\tan x)\,dx$$ I think the integral is quite simple but I am having a hard time evaluating it. I started with the result: $$\int_0^{\pi/4} \ln(\tan ...
0
votes
1answer
35 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
1
vote
3answers
51 views

The value of $\int_0^{2\pi}\cos^{2n}(x)$ and its limit as $n\to\infty$

Calculate $I_{n}=\int\limits_{0}^{2\pi} \cos^{2n}(x)\,{\rm d}x$ and show that $\lim_{n\rightarrow \infty} I_{n}=0$ Should I separate $\cos^{2n}$ or I should try express it in Fourier series?
7
votes
2answers
95 views

Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$

At first sight it looks like the integral below $$\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$$ can be evaluated by using some geometric series. What else can we do? Is there a fast easy ...
1
vote
0answers
10 views

Solving ODE involving matrices

We have a given ODE $ K(x)_{_{3 \times 3}}=xC_1K(x)+x^3C_2K'(x) \tag 1$ where $C_1,C_2$ are constant skew symmetric matrices of dimension $3 \times 3$ with determinant $0$. How do we solve ...
2
votes
0answers
37 views

Time to buy a house without a mortgage equation!!

I am looking into a "real world" calculation to calculate the time taken for someone to buy their own home while they rent it. They do this by buying small pieces of the property every month, and ...
1
vote
1answer
24 views

Error function - Not seeming to come out right

I have reached two integrals: $$\int_{(x+L)/(2c\sqrt{t})}^\infty e^{-z^2} dz + \int_{-\infty}^{(x-L)/(2c\sqrt{t})} e^{-z^2} dz$$ Now the first evaluates just fine to ...
2
votes
1answer
48 views

Setting up integrals for the moments and the center of mass of a planar region

Problem Consider the following region: a semi-circle with radius = 3 ft on top of a rectangle with height = 11. (with constant density) a.) Set up integrals for the moments, Mx, My, and the center ...
3
votes
2answers
189 views

Finding the Integral of a function

$\int \dfrac{1}{9+x^2}dx$ The answer is $\arctan(x/3)/3 + C$ , but I don't understand the process of how the answer was found. I tried using u-substitution, but I came up with 2xdx/x^2+9.
5
votes
0answers
46 views

Closed form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

We know a closed-form of the first two powers of the integral of trilogarithm function between $0$ and $1$. From the result here we know that $$I_1=\int_0^1 \operatorname{Li}_3(x)\,dx = ...
0
votes
1answer
16 views

Solving a separable differential equation: What's wrong with my calculation?

Solve the following separable differential equation:$$\frac{dy}{dx}=(-4)\cdot e^y \cdot cos(4x)$$ My answer (which is incorrect but I don't know why): $$\frac{dy}{dx}=(-4)\cdot e^y \cdot cos(4x)$$ ...
3
votes
2answers
220 views

Calculating integral of function with square root

I don't have any idea how to calculate this integral. I learned to calculate the elementary functions, and how to calculate by the positioning way. For example $t=e^x+1 \rightarrow dt=e^x \, dx$ and ...
3
votes
1answer
52 views

Is this an acceptable way to integrate?

I am supposed to find: $$ \int \sec(1-x)\tan(1-x) dx $$ I then set $ u = \sec(1-x) $ $$ du = -\tan(1-x)\sec(1-x)\ dx $$ therefore $$ \frac{-du}{\sec(1-x)} = \tan(1-x)\ dx$$ Which when applied gives ...
1
vote
2answers
93 views

Factorial Rational Limit

Anything besides the squeeze theorem. Here it is: $$\lim_{n\to\infty} \frac{(2n - 1)!}{{2n}^{n}}$$ Can someone start me off?
2
votes
1answer
20 views

Finding Factorial using Integral Definition

$n! = \int_{0}^{\infty} {e}^{-x}{x}^{n} \,dx$ How can we find $400!$? $400! = \int_{0}^{\infty} {e}^{-x}{x}^{400} \,dx$ Integration by parts is way too complicated, what are the other options?
1
vote
1answer
32 views

Find the area bounded by the given curves

x+y=3 and the coordinate axes. I know how to find the area bounded by 2 curves it's just that I'm confused with "coordinate axes". Is it the same as x=y? or not? please help me understand. okay, ...
0
votes
0answers
9 views

Existence and uniqueness of Volterra integral equations of the first kind with vanished kernel

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...
11
votes
2answers
117 views

How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$

The below problem is taken from Joseph Edwards book Integral Calculus for beginners. How does one show: $$5 \int \frac{\sin(x)}{\sin(5x)} \ dx= \sin\left(\frac{2\pi}{5}\right) \cdot ...
0
votes
1answer
20 views

Darboux integrals with bisected partition

Let us call $\overline{\int_a^b}f(x)dx$ the Darboux upper integral of $f$ and $\underline{\int_a^b}f(x)dx$ the lower one. Let us construct a partition of $[a,b]$ into $2^n$ intervals $[x_{k-1},x_k]$ ...
5
votes
2answers
69 views

Is this closed form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?

According to Freitas' paper at page $11$. $$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)-\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$ I evaluated the LHS and it is ...
0
votes
2answers
25 views

Calculating a calculable Integral using Integration

I am having trouble with an integral: $$\frac2L\int_L^\infty C\sin\left(\frac{n\pi x}{L}\right) dx$$ Where $C$ is just a constant. I can't see how to do this, despite it apparently being rather ...
0
votes
0answers
3 views

find the kernel of voltera 2nd kind with particular form 2. (Alternative approach)

find the kernel of voltera 2nd kind with particular form 2. (Alternative approach) in which we take kernel function of x and t ot just x or just function of t. we try to solve it with alternative ...
5
votes
3answers
117 views

Some integral representations of the Euler–Mascheroni constant

What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\infty ...
3
votes
3answers
45 views

Using $x=\tan \theta$ to solve $\int x\sqrt{1+x^2}\,\mathrm dx$

I'm having a lot, I repeat, a lot of trouble with Calculus II, particularly trigonometric substitution. At the moment, I'm extremely confused as to how to integrate $\int x\sqrt{1+x^2}\,\mathrm dx$ ...
0
votes
1answer
68 views

Triple integration and transformations [on hold]

Hello I am a 17 year old kid in high school that does math for fun. One of my buddies gave me these problems and I can't seem to figure it out. Question 1 Find the volume of the finite region in ...
0
votes
0answers
22 views

Integration to find a value between a constant and infinity

I don't have any idea how to go about solving this so hints are preferred. Let f(x) be a continuous function defined on the interval $[2, \infty )$ such that $f(4) = 15$, $|f(x)| < x^{7} + 11$, ...
5
votes
1answer
81 views

Integral ${\large\int}_0^1\frac{\ln^2\ln\left(\frac1x\right)}{1+x+x^2}dx$

Gradshteyn & Ryzhik, 7th ed., p. 570, formula 4.325(5) give the following definite integral: ...
0
votes
1answer
38 views

How can the integral of $|\sin(x)|$ be $-\cos(x)\text{sgn}(\sin(x))$?

Wolfram|Alpha tells me that $\int|\sin(x)| = -\cos(x)\text{sgn}(\sin(x))$ (which happens to also be its derivative), but I don't understand how this is possible, because the resulting function jumps ...
0
votes
1answer
18 views

Solve the integral with complex number and floor function.

let $z\in\mathbb{C},\,0<\left|z\right|<1$.I would like to calculate the integral $$\int_{1}^{\infty}\frac{\left(1-z\right)^{\left\lfloor t\right\rfloor }}{t^{2}}dt$$ where ${\left\lfloor ...
1
vote
2answers
36 views

Area between $y=x^4$ and $y=x$

The problem I'm having some trouble solving is this: calculate the area between $y=x^4$ and $y=x$. The points are $a = 0$ and $b = 1$, but the definite integral is negative. What am I doing wrong ...
3
votes
0answers
42 views

How to evaluate the following integrals

$$\int\limits_0^{\frac{\pi }{2}} {{x^2}{{\ln }^2}\left( {\sin x} \right)\ln \left( {\cos x} \right)dx} ,\int\limits_0^{\frac{\pi }{2}} {x\ln \left( {\sin x} \right){{\ln }^2}\left( {\cos x} \right)dx} ...
0
votes
1answer
20 views

calculus integration, average height of point on semi circle

i was recently watching a single variable calculus video of mit 18.01, lecture 23. in that it is said that average height of a point on semicircle with respect to arc length is 2/pi.I have a hard time ...
0
votes
1answer
38 views

What can be said about the inverse of the antiderivative of a strictly positive function?

Let $f:\mathbb R\rightarrow [1,\infty)$ be (a strictly positive) function. Define $$F(t) = \int_0^t f(s)ds.$$ Obviously, $F$ is injective and hence invertible. How does $F^{-1}$ look like?
3
votes
1answer
112 views

Closed form of $\int_0^{\pi/4} \sin(\sin(x)) \, dx$

Let $I(b)$ is the following integral $$I(b)=\int_0^b \sin(\sin(x)) \, dx.$$ There are some $b$ value for that we know a closed-form of $I(b)$ in term of Struve function $\mathbf{H}_n(x)$. For ...
1
vote
0answers
26 views

exact angle between the functions $p(x)=3x-4$ and $q(x)=9x-5$ over $0\leq x\leq 1$

I was attempting a question, which gave the formula for the angle theta between two functions $f(x)$ and $g(x)$ over $a\leq x\leq b$ (note the question defined the meaning of norm and inner product as ...
0
votes
1answer
21 views

Average value of $f(x,y,z)$ over region $W$

Can anyone show me how to do this problem? Compute the average value of $f(x,y,z)$ over the region $W$. $f(x,y,z) = xyz$ $W$ : $0 \le z \le y \le x \le 1$ Thank you.
0
votes
1answer
52 views

Integral solution / simplification

I am encountering difficulty in solving the following integral: $$\int_{0}^\infty(1-x)^{a_2-1}x^{a_1-1}dx$$ Could you suggest a substitution using the sum of an infinite geometric/Taylor or other ...
1
vote
1answer
29 views

Double Integration Inequality

I've been trying to work out the following. Could anyone please show me how to do this? Let $D$ be the domain bounded by $y=x^2+1$ and $y=2$. Prove the inequality $$\frac{4}{3}\le ...
0
votes
0answers
13 views

Changing a double integral to single integral

I have seen this integration problem in a random process text book. We have the following integral. $\int_{-T}^{T}\int_{-T}^{T}C(t_1-t_2)dt_1dt_2 = \int_{-2T}^{2T}(2T-|\tau|)C(\tau)d\tau$ where ...
19
votes
1answer
800 views

The Wicked Integral

My brother's friend gave me the following wicked integral with a beautiful result \begin{equation} {\Large\int_0^\infty} \frac{dx}{\sqrt{x} \bigg[x^2+\left(1+2\sqrt{2}\right)x+1\bigg] ...
3
votes
3answers
104 views

Integration: $\int_0^{ + \infty } \frac{\cos x}{x} dx$

Although I have known that $\displaystyle\int_0^{ + \infty } {{\sin x} \over x} \, dx = {\pi \over 2}$, I have no idea how to work out $\displaystyle\int_0^{ + \infty } {{\cos x} \over x} \, dx$. How ...
1
vote
2answers
87 views

Solving ODE containing matrices

We have an ODE $ \psi'(t)_{_{3 \times 3}}=\psi(t)_{3 \times 3}(A_{3 \times 3}+B_{3 \times 3}t)\tag 1$ Given Data in Question We have no quarentee that $\psi'(t),\psi(t)$ both have inverse A,B are ...
7
votes
4answers
123 views

Integrate $\int\frac{dx}{(x^2+1)\sqrt{x^2+2}}$

I would like some guidance regarding the following integral: $$\int\frac{dx}{(x^2+1)\sqrt{x^2+2}}$$ EDIT: The upper problem was derived from the following integral ...
0
votes
3answers
38 views

How to $\int 3t \sqrt {t^2+1} dt$

$$\int 3t \sqrt {t^2+1} dt$$ My book has done $$\int_0^2 3t (1 + t^2)^{\frac{1}{2}}dt = \left[(1 + t^2)^{\frac{3}{2}}\right]_0^2.$$ They have done so simple way, but how is this correct? I have ...
0
votes
1answer
15 views

Cauchy Principal Value different from the improper integral

I am having trouble formulating an example for which $\mathcal{P}\int^{\infty}_{-\infty}f(x)dx\neq\int^{\infty}_{-\infty}f(x)dx$ Would an example be $f(x)=1/x$ because of the asymptote at $x=0$?