0
votes
0answers
20 views

area of surface Function for Champagne flute

Can any body tell me how to solve number 5 on this page i have attached, finding a function for champagne flute while function for wine glass is given.
0
votes
4answers
78 views

Why is $\int_{0}^{2\pi} |\sin x| dx = 4$

I can't understand why $$\int_{0}^{2\pi} |\sin x| dx = 4$$while $$\int_{0}^{2\pi} \sin x dx = 0$$ I did the calculus for the second varian but I can't reach result $4$ for the first integral. Thank ...
1
vote
1answer
38 views

Hairy $u$-substitution problem

Evaluate using the $u$-substitution: $$\int_{-1}^{1} \frac{dx}{4 + x^2}.$$ Now, I was told to set $$\tan u = \frac{x}{2},$$ but that doesn't help me at all. Hints needed!
2
votes
1answer
55 views

definite integral involing hyperbolic and trigonometric functions

Trying to prove the following: $$ \int_0^\infty xe^{-c x^2}\sinh(a x)\cos(bx)\,dx = ...
2
votes
4answers
82 views

Solving the integral $\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$

I really don't know how to solve this integral $$\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$$ Should I use firstly a formula of $\sin(a+b)$?
1
vote
1answer
70 views

Evaluation of the integral $\int_{-6}^{-3}\frac{\sqrt{x^2-9}}{x}$

How to evaluate the following integral? I have tried the following things but I have no idea to continue after the last step. Moverover, the integral seems wrong when compared with the ans from ...
0
votes
1answer
19 views

Trigonometric Integral of variable function.

Let for any $n \in \mathbb Z$, define a function $f_n \text { on } [0,1]$ as follows: $$f_n(x) = \begin{cases}0 &\text{if} &x=0 \\ \sin ...
1
vote
1answer
35 views

A trigo integration

How to evaluate $\displaystyle\int_{-1}^{1}\frac{1}{1-e^x\sqrt{1-x^2}}dx$ I have to tried $x=\sin t$ or $\cos t$ but I can't solve $\displaystyle\int_{0}^{\pi}\frac{1}{1-e^{\cos t}}dt$ can anyone ...
1
vote
3answers
81 views

Integral of $\int^\sqrt2_1\frac{1}{1+\sqrt{x^2 - 1}}dx$ by substitution?

In a maths question I have the question: $$\int^\sqrt2_1\frac{1}{1+\sqrt{x^2 - 1}}dx$$ by substitution? All other questions have been by trigonometric substitution so I assume that is how to solve. ...
0
votes
1answer
38 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
2
votes
2answers
93 views

Integral of complex questions?

$$\int_0^{\pi/4} \frac {\sin x + \cos x}{\sin^4x+\cos^2x}dx$$ $$\int e^x\cot x(\csc x-1)dx$$ These two integrals are impossible to find. If anyone knows how to integrate them please help me. I am ...
3
votes
5answers
204 views

How to evaluate this Trig integral?

I need to find the definite integral of $\int(1+x^2)^{-4}~dx$ from $0$ to $\infty$ . I rewrite this as $\dfrac{1}{(1+x^2)^4}$ . The $\dfrac{1}{1+x^2}$ part, from $0$ to $\infty$ , seems easy ...
3
votes
2answers
541 views

Definite integral involving trigonometric functions and absolute values

Solve the following integral: $$ \int_{0}^{4\pi}\frac{x|\sin x|dx}{1 + |\cos x|} $$ I tried variable substitution, but nothing seemed to work. Could you give me some clues?
-1
votes
2answers
61 views

Evaluate $\int_o^{\pi ^2\over4}\cos^2(\sqrt{x})\, \operatorname d\!x$?

I want to integrate following: $$\int_o^{\pi ^2\over4}\cos^2(\sqrt{x})\,dx.$$ I try solve instead: $$\int_o^{\pi ^2\over4}\frac{\cos(2\sqrt{x})+1}{2} \,dx.$$ but I can't integrate $\cos(2\sqrt x)$
1
vote
0answers
46 views

An other tricky one. Tigonometric integral.

How should I attack this ? $$ \int_0^{\pi} \cos(ax)^m \cos(x)^n dx$$
2
votes
1answer
77 views

Tricky trignometric integral. Any ideas?

I have been scratching my head with this integral. Any ideas that I could try? $$ \int_0^{\pi}\sin(\theta) \cos(\theta)^{n+m}\left(1-a\tan(\theta)\right)^m d\theta$$
0
votes
0answers
38 views

integral 2D involving complex exponential and cosine

I've some doubts about my solution of this integral: $$I(\phi_{1},\phi_{2})=\int_0^ {2\pi} \,d\phi_{1}\int_0^ {2\pi} \,d\phi_{2} \frac{e^{-in\phi_{1}} e^{-im\phi_{2}}}{2\pi}\frac{e^{il\phi_{1}} ...
0
votes
2answers
127 views

Derivative of integral of $\sin (t^2)$

I'm stuck with the problem If $ F(x)=\int_0^{x^3} \sin t^2 dt$ find $F'(x)$ Now, if the upper interval were $x$, the answer would be $\sin t^2$ (right?). However, the upper interval is $x^3$. ...
1
vote
3answers
83 views

$W_n=\int_0^{\pi/2}\sin^n(x)\,dx$ Find a relation between $W_{n+2}$ and $W_n$

Set $$W_n=\int_0^{\pi/2}\sin^n(x)\,dx.$$ Compute $W_0$ and $W_1$. Find a relation between $W_n$ and $W_{n+2}$ and deduce a formula for $W_n$. What I have so far is: $$W_{2k}=\frac{1}{2^k}\left( ...
8
votes
3answers
110 views

How to demonstrate the equality of these integral representations of $\pi$?

Each of the following definite integrals are well known to have the value $\pi$: $$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}dx=2\int_{-1}^1\sqrt{1-x^2}dx=\int_{-\infty}^{\infty}\frac{1}{1+x^2}dx=\pi.$$ I ...
13
votes
2answers
399 views

Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$

Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
5
votes
2answers
68 views

A trigonometic integral with complex technicals

Let $a,b \in \mathbb{R}^+$. Show that $$\int_0^{2\pi}\frac{1}{a^2\cos^2(t)+b^2\sin^2(t)}dt=\frac{2\pi}{ab}$$ Help please! Thanks.
1
vote
0answers
147 views

Definite integral involving exponential, powers and trigonometric functions

Is it possible to evaluate the following integral? $$ \int_{-\pi}^{\pi} e^{-qx^{ak}(x^2+d^2+2 d x Cos[t])^{-a/2}} dt $$ I am not able to find any related formula. Note that this integral follows ...
1
vote
0answers
57 views

Definite integral involving powers and trigonometric functions

Is it possible to evaluate the following integral? $$ \int_{-\pi}^{\pi} \frac{m}{m+x^{ak}(x^2+d^2+2 d x Cos[t])^{-a/2}} dt $$ I am not able to find any related formula, while I think Mathematica ...
27
votes
2answers
700 views

Integral $\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)dx$$
25
votes
2answers
1k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
16
votes
2answers
661 views

Integral $\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$

Please help me to evaluate this integral: $$\large\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$$
6
votes
6answers
813 views

Proof that $\int_0^{2\pi}\sin nx\,dx=\int_0^{2\pi}\cos nx\,dx=0$

Prove that $\int_0^{2\pi}\sin nx\,dx=\int_0^{2\pi}\cos nx\,dx=0$ for all integers $n \neq 0$. I think I'm encouraged to prove this by induction (but a simpler method would probably work, too). ...
9
votes
1answer
132 views

An integral involving the inverse of $f(x)=\log x-\log\cos x+x\tan x$

Let the function $f:\left(0,\,\displaystyle\frac\pi2\right)\to\mathbb{R}$ be defined as $$f(x)=\log x-\log\cos x+x\tan x.$$ Let its inverse be denoted as ...
1
vote
2answers
130 views

Solve the following definite integral

Solve the following integral: $$\int_{0}^{∞}\frac{x^2dx}{({1-x^2})^2}$$ I know that substituting some trigonometric functions may help. But I was not able to solve. Can you give me some ...
19
votes
1answer
566 views

A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

I need to find a value of this definite integral: $$\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx.$$ Its numeric value is approximately $0.7875720991394284$, and lookups ...
1
vote
2answers
55 views

Dilemma evaluating integral after sinusoidal substitution

To solve the definite integral $$ I = \int_{-a}^{a} \frac{dx}{\pi \sqrt{a^2-x^2}}$$ I used the substitution $x = a \sin \theta$ and tried to solve the integral without its interval definition, ...
4
votes
1answer
213 views

Integrating $\int_0^{\pi/2} \cos^a(x) \cos(bx) \ dx$

Please help me in this integral : $$\int_0^{\pi/2} \cos^a(x) \cos(bx) \ dx \quad \text{if}\; b>a>-1$$ Please help me I used everything and can't evaluate it.
4
votes
0answers
78 views

Generalized sine integral $ \int_0^\infty \sin^m(x^n)/x^p\,\mathrm dx$

I have seen that both the integrals $$ \color{black}{ \int_0^\infty \operatorname{sinc}(x^n)\,\mathrm{d}x = \frac{1}{n-1} \cos\left( \frac{\pi}{2n}\right)\Gamma ...
4
votes
2answers
116 views

Integrating $e^{ \cos(x)\sin(x)}$ from $0$ to $2\pi$

So i'm trying to find out how, or if its even possible, to integrate $e^{\sin(x)\cos(x)}$ analytically from $0$ to $2\pi$. I know that i can integrate $e^{\cos(x)+\sin(x)}$ or $e^{\cos(x)^2}$ and ...
2
votes
2answers
94 views

How can this property of definite integrals be true?

In this question, people are saying that the definite integral of $f(x)$ from $0$ to $a$ is equal to the integral of $f(a-x)$ from $0$ to $a$. How can that be true? Simple examples don't work.
2
votes
4answers
84 views

Definite integration of a trigonometric function

How to integrate $$\int_0^{\pi/2}\!\dfrac{2a \sin^2 x}{a^2 \sin^2 x +b^2 \cos^2 x}\,dx $$ my first step is $$\frac{2}{a} \int_0^{\pi/2}\!\dfrac{a^2 \sin^2 x}{a^2 +(b^2 - a^2) \cos^2 x}\, dx $$ I ...
16
votes
1answer
252 views

$\int_0^\infty\text{Ci}(x)^3\mathrm dx$

Is there a closed form for this integral: $$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$ where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
26
votes
3answers
696 views

$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$

Please help me to solve this integral: $$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$ I managed to calculate an indefinite integral of the left part: $$\int\frac{\cos x}{\sin ...
3
votes
1answer
47 views

Using a particular image to justify a (specific) trig integral equality.

I would like to include the following string of equalities in a paper: $$\sin ^2(x) + \cos ^2(x) = 1$$ $$\int _0^{\dfrac{\pi}{2}} \sin ^2 (x)dx + \int_0^{\dfrac{\pi}{2}} \cos ^2 (x)dx = ...
1
vote
2answers
366 views

Evaluating a double integral involving exponential of trigonometric functions

I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ ...
32
votes
2answers
697 views

Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$

Prove that $$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
2
votes
1answer
194 views

Definite double integral of a trigonometric function

I am in trouble in the calculation of the following double integral: $$\int_0^a d\rho\int_0^{2\pi}d\phi\exp(-ik\rho(\sin(\theta_0)\cos(\phi_0-\phi)+\sin(\theta_1)\cos(\phi_1-\phi)))\rho$$ Many thanks
2
votes
3answers
151 views

Finding integral $\int_{- \pi}^{\pi} \cos(mt) \cos(\lambda t) dt$

I am a little stuck on the following problem: Prove that: $$\int_{- \pi}^{\pi} \cos(mt) \cos(\lambda t) dt = -2 \frac{(-1)^{m} \lambda \sin(\pi \lambda)}{m^2 - \lambda ^2}$$ I have used the fact ...
0
votes
1answer
75 views

Is this conjecture about integration of sinusoids on a specific interval correct?

I haven't formally learned integrals, but I was trying to apply what I do know. Is $$\int_{a}^{b}m\sin(k(x+j)) dx =0$$ as long as $b-a\equiv 0 \pmod {\frac{2\pi}{k}-j}$ and $m, k, j \in ...
5
votes
1answer
141 views

How to evaluate $\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$?

How can I integrate the following: $$\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$$ for $a,b>0$? Maple gives a compact result: $$\frac{1}{b} \tan^{-1}(c) - \frac{1}{ac^2} ...
3
votes
1answer
140 views

evaluation of the integral $\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $

Can the integral $$\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $$ be expressed in terms of elemental functions or in terms of the sine and cosine integrals ? if possible i would need a hint thanks. ...
4
votes
1answer
244 views

How can I compute $\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$

If $f(x)=\text{arccot}(x)$ for non-negative $x$ and $0$ otherwise, how can I calculate $$\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$$ for $y\in\mathbb{R}$?
2
votes
1answer
172 views

Definite integral with a complex number in Euler form

Well... I spent an hour trying to figure out how to go from lhs to rhs: $$\frac { 1 }{ 2\pi } \int _{ -\infty }^{ +\infty } \phi _{ T }(u)\left( \int _{ k }^{ +\infty } e^{ -iux }dx \right) ...
6
votes
5answers
412 views

A little integration paradox

The following integral can be obtained using the online Wolfram integrator $$ \int \frac{dx}{1+\cos^2 x} = \frac{\tan^{-1}(\frac{\tan x}{\sqrt{2}})}{\sqrt{2}}$$ Now assume we are performing this ...