Questions tagged [trigonometric-integrals]
Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.
827
questions
3
votes
1answer
53 views
Help on how to proceed on this trig integral
i would appreciate if you could help me with this problem.
$$I=\int_{0}^{\pi}\frac{x \sin^{2018}(x)}{\cos^{2018}(x)+\sin^{2018}(x)}dx$$
I am completely overwhelmed on how to proceed with this and i am ...
1
vote
1answer
79 views
Tips on how to start calculating a definite integral
i would appreciate if you could help me with this problem.
I don't want the answer only how should i proceed with solving an integral like this
$$I=\int_{0}^{\pi}\frac{x \sin^{2018}(x)}{\cos^{2018}(x)+...
2
votes
1answer
36 views
On using sine and cosine substitutions in integration
When you make a substitution of the form $x=asin\theta$ for sines and cosines, shouldn't you check if the values of x on which we're integrating allows for it? Otherwise, if x becomes greater than |x|,...
2
votes
1answer
41 views
Solving a nonlinear first order ODE
In this text https://www.springer.com/gp/book/9781461454762 on p. 95, it has the following
The text does not explain how to get to this solution. $B, C, D$ are arbitrary constants. With $A>0$, I ...
0
votes
2answers
37 views
Nonlinear System of ODES
I have been trying to solve this nonlinear system of ODEs analytically with no luck:
$$\frac{d\psi}{dt}=-cot \theta cos \psi$$
$$\frac{d\theta}{dt}=-sin\psi$$
$$\frac{d\phi}{dt}=\frac{cos\psi}{sin\...
0
votes
2answers
32 views
Evaluate the integral using Trigonometric Substitution [closed]
$$\int_{0}^{3}\frac{x^3}{(3+x^2)^{\frac{5}{2}}}dx$$
-2
votes
0answers
28 views
Use Trigonometric Substitution [closed]
Evaluate the integral using Trigonometric Substitution
\begin{equation*}
\int ^{2}_{\sqrt{2}}\frac{\sqrt{2x^{2} -4}}{x} dx
\end{equation*}
0
votes
2answers
82 views
About $\int_{-\infty}^{\infty} \frac{\sin \left( x\right )}{x} \mathrm{d}x = \int_{-\infty}^{\infty} \frac{\sin ^ 2\left( x\right )}{x^2} \mathrm{d}x$
How to show $\displaystyle\int_{-\infty}^{\infty} \frac{\sin \left( x\right )}{x} \mathrm{d}x = \int_{-\infty}^{\infty} \frac{\sin ^ 2\left( x\right )}{x^2} \mathrm{d}x$ ?
I've found this identity in ...
8
votes
1answer
203 views
Find $ \lim_{r \to 1^{-}} \int_{-\pi}^{\pi} (\frac{1+2r^2}{1-r^2\cos2\theta})^{1/3}d\theta$
Evaluate $$ \lim_{r \to 1^{-}} \int_{-\pi}^{\pi} \left[\frac{1+2r^2}{1-r^2\cos\left(2\theta\right)}\right]^{1/3}{\rm d}\theta $$
Question- Can I take the limit inside the integral?
My try-
$$I= \lim_{...
1
vote
1answer
43 views
A problem with Weirstrass substitution
I hope you all are doing well.
Yesterday, I was doing some undefined integral exercises when I faced an exercise that was something like:
$$\int \frac{F(\sin(x),\cos(x))}{G(\sin(x), \cos(x))}dx$$
I ...
2
votes
1answer
60 views
Need help with proof on trigonometric integrals
Question:
Let $f:[0,\pi]\rightarrow\mathbb{R}$ be a continuous function. Show that, if $$\int\limits_0^{\pi} f(t)\sin(t)dt = \int\limits_0^{\pi} f(t)\cos(t)dt =0,$$ then the equation $f(x)=0$ admits ...
0
votes
2answers
49 views
How do I evaluate $\int_{0}^{\sqrt{2}}\int_{x}^{\sqrt{4-x^2}}{\sqrt{x^2+y^2}} \, dy \, dx$?
I'm having troubles evaluating this double integral. Can somebody help me? I've gone to the part that I need to use trigonometric substitution, but performing the said sub, I think I'm kind of unsure ...
0
votes
0answers
37 views
How to do a very complicated integral involving trig functions?
I am trying to do this complicated integral:
$$\int\limits_0^{2\pi}\frac{(t-c-\pi)(2 \cos(t-b) \sin(t-b) \cos(t-c) \sin(t-c)-2 \sin(t-b) \sin(t-c)+2 \cos(t-b)^2 \cos(t-c)^2-(\cos(t-c)+\cos(t-b))^2+2)}{...
4
votes
2answers
153 views
$\int_0^{2 \pi} \sinh^{-1} \frac{h}{\sqrt{s^2 + R^2 -2sR \cos(\phi)}} \, d\phi $
The definite integral is the following one:
$$I_0 = \int_0^{2 \pi} \sinh^{-1} \frac{h}{\sqrt{s^2 + R^2 -2sR \cos(\phi)}} \, d \phi $$
Where $h, s, R$ are positive real quantities and $\sinh^{-1}$ is ...
-1
votes
1answer
51 views
Why does my answer to the integral of $\frac{x^2}{x^2+9}$ need to be multiplied by $3$
This is my work below, the answer to the problem was $3$ times what my solution was. Why?
The integral was the indefinite integral of $\displaystyle \int\frac{x^2}{x^2+9}$ and my answer was $\frac{x}{...
0
votes
1answer
42 views
Integration by inverse trigonometric substitution
I'm trying to integrate by inverse trigonometric substitution, and I have an answer (though I'm pretty sure it is wrong), but it is not the answer that the solution provides. I need to check where I ...
3
votes
3answers
73 views
Help with inverse trigonometric substitutions $ \int x^2\sqrt{a^2+x^2}\,dx $.
how would I go about integrating this? It is a lecture exercise.
$$
\int x^2\sqrt{a^2+x^2}\,dx
$$
I used a substitution of $x=a\tan\theta$ and ended up with $$\int a^4\tan^2\theta \sec^3\theta \,d\...
1
vote
1answer
45 views
To find an integral that satisfies a substitution property.
Consider the following integral. $$\int \cfrac{1}{9-x^2}dx$$
Substituting $x=3\sin u$ makes the integral $\displaystyle\int\cfrac{1}{3\cos u}du$.
I would like to modify the original integral such that ...
-1
votes
1answer
61 views
Show that $ \tan x +\cot x = \frac{2}{ \sin 2x }$ [closed]
This is a two part question I have no idea how to do this I stugle with prove/show that question and my book only says proof so I don't know how to get to it.I appricat it. Thank you!
The second pat ...
1
vote
1answer
74 views
Find the value: $\int_{0}^{Ļ/6}\ 4\sin^{2}xdx$
Solving: $\int_{0}^{\frac{\pi}{6}} 4\sin^2(x)dx$
This is my work
but I can not seem to get to the answer $\frac{1}{6}(2\pi-3\sqrt{3})$. Any help is much appreciated. Thank you!
0
votes
2answers
45 views
Prove $\cos(2x)-\cos(4x)=2\sin(3x)\sin(x)$
Use the expansion of $\cos(3x-x)$ and $\cos(3x+x)$ to show that $\cos(2x)-\cos(4x)=2\sin(3x)\sin(x)$
I am so confused. I get that $\cos(2x)-\cos(4x)=\cos(3x-x)-\cos(3x+x)$
This part I don't get; the ...
0
votes
0answers
60 views
Fourier Sine transform of $\arctan(x/a)$.
How can I find the Fourier sine transform of $arctan(x/a) ;a>0$? I am solving it as,
$$F_{s}\left[\arctan\left(\frac{x}{a}\right)\right]=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\arctan\left(\frac{x}{...
0
votes
2answers
31 views
How to rearrange the following trigonometric integral
How do you rearrange the following;
$\frac{1}{R_0}\int_{0}^{\pi/2}\frac{1-sin^3(\alpha)}{cos^2(\alpha)} d\alpha$
using t=$\tan\frac{\alpha}{2}$
to obtain
$\frac{1}{R_0}\int_{0}^{1}[\frac{2}{(1+t)^2}+4\...
0
votes
2answers
92 views
Show that $\int_1^\infty x^n\sin(x+x^2)dx$ diverges, $n\ge1$.
How do I show that this improper integral diverges? I know certainly it is not always true that if $\int_a^\infty f(x)dx$ converges, then $f(x)\to 0$. My intuition tells me this improper integral ...
2
votes
0answers
17 views
What's wrong with my approach to solving this integral via trigonometric substitution?
(I'm sorry, I've spent the last hour trying to figure out Latex, but nothing's working. I'm just going to paste my work as I've written it in pencil.)
I'm trying to solve this equation:
Integral from ...
0
votes
1answer
27 views
How can i calculate the contour integral of this trigonometric function?
Here's my integral $\int_0^{2\pi} \frac{cos(n\theta)}{cosha+acos\theta}d\theta$ where $|a|<1$
I tried $\int_0^{2\pi} \frac{e^{in\theta}}{cosha+acos\theta}d\theta$ but stuck here because of ...
2
votes
2answers
43 views
$\int_{0}^{r} \sqrt{\frac{1}{r^2-x^2}} \ dx$ with improper integration or trigonometric substitution?
I've always done this limit with improper integration because the denominator goes to zero, but is a trigonometric substitution valid as well? (edit, I know of the trigonometric substitution $x=r \sin\...
1
vote
0answers
23 views
Indefinite integral resulting in inverse trigonometric functions
I tried to find an answer to this question but found nothing, maybe because I don't know how to properly formulate it. I hope I can make myself understood here.
The thing is, recently I started ...
2
votes
3answers
91 views
Solve integral $\int\frac{1}{(x^2-1)\sqrt{x^2+1}}dx$
$$\int\frac{1}{(x^2-1)\sqrt{x^2+1}}dx$$
I'm trying to solve this integral. First I substituted : $x=\tan(t)$; $t=\arctan(x)$
Then $$ dx=\frac{1}{\cos^2(t)}\,dt$$
Now by simplifying I'm to this step $$ ...
2
votes
2answers
48 views
Solve $\lim\limits_{x\to\infty}{\frac{1}{\sqrt{x^2-1}}\int_{0}^{x}{{\left(\arctan{t}\right)}^{2}\mathrm{d}t}}$
I have to solve the following limit: $\lim\limits_{x\to\infty}{\frac{1}{\sqrt{x^2-1}}\int_{0}^{x}{{\left(\arctan{t}\right)}^{2}\mathrm{d}t}}$. The problem for me is the definite integral. I tried $t=\...
2
votes
2answers
39 views
How to write $\sin(x_1).\sin(x_2)$ as an integral?
We know that $\sin(x)$ is the same as writing $\int_{0}^3 \cos(x)dx$.
However, I am interested if there is a similar way to write $\sin(3). \sin(4) $ as an integral.
One way I think is $$\sin(3).\sin(...
4
votes
2answers
71 views
How to evaluate $ \int_{0}^{\pi/2} \arctan(\sin(x)) \,dx + \int_{0}^{\pi/4} \arcsin(\tan(x)) \,dx$?
Consider the integral $$ \int_{0}^{\pi/2} \arctan(\sin(x)) \,dx + \int_{0}^{\pi/4} \arcsin(\tan(x))
\,dx$$
I tried using this substitution for the first integral
$$\arctan(\sin(x))=t \rightarrow x= \...
1
vote
2answers
90 views
Integrating $\int_0^\infty \frac{\sin (ax)}{x^3}dx$
I am trying to integrate this integral,
$$I_3 = \frac{ -4 }{ \pi} \int_0 ^{\infty} \frac{d\lambda}{\lambda^4} \cdot \sin(p_1 \lambda)\cdot \sin(p_2 \lambda) \left(-p_3 \lambda\cos\left(p_3 \lambda\...
0
votes
1answer
41 views
Eulerās gamma functions and exponential integrals
I am trying to solve these integrals,
where $ (C_i,S_i) ā” (\cos Ī»p_i, \sin Ī»p_i)$
I have done $D_{SS}$. But in $D_{SC}$, I encounter some divergences.
The author says this in the paper:
The ...
2
votes
0answers
54 views
$\lim_{x\to a}\frac{a}{x-a}\operatorname{Si}_a(x)$
Let $a>0$ and $\operatorname{Si}_a(x)=\int_{a}^{x}\frac{\sin(t)}{t} \, dt$. Compute
\begin{equation*}
\lim_{x\to a}\frac{a}{x-a}\operatorname{Si}_a(x)
\end{equation*}
My reasoning was: suppose $F$ ...
1
vote
2answers
103 views
Integration with $\int_0^\infty \frac{\cos (ax)}{x^2}dx$
I am trying to integrate $$\int_0^\infty \frac{\cos (ax)}{x^2}dx$$
I get $$-a\operatorname{Si}(ax)-\frac{\cos(ax)}{x}+C$$ as indefinite integral and when I put limits, I understand how to get $-\frac{{...
0
votes
1answer
50 views
Using residue theorem to evaluate integral and calculating residues.
Evaluate the integral $\int_{0}^{2\pi} \frac {\cos^2(x)}{13+12\cos(x)} \,dx$ using the residue theorem.
I have managed to make a start on this problem by putting this problem in a complex analysis ...
0
votes
0answers
29 views
Convolution - when do you use integration limits of infinity and when you use 0 to t
So I have been taught the definition of convolution where the limits go from minus infinity to positive infinity.
Currently I am trying to solve
$f(t) \cdot g(t)$
Where f(t) = t
And g(t) = cos3t
When ...
2
votes
2answers
46 views
If $p$ and $q$ are solutions of the equation $x \tan x = 1$, show the integral of $\cos^2 px$ entirely in terms of $p$
I am working through a pure maths text book out of interest. I have finished the chapter on integration and differentiation of trigonometric functions and am doing the end of chapter questions. This ...
4
votes
3answers
85 views
$\int x^{2}\sqrt{a^{2}+x^{2}}\,dx$. Is there another way to solve it faster?
I have to calculate this integral:
\begin{align} \int x^{2}\sqrt{a^{2}+x^{2}}\,dx \qquad\text{with}
\quad a \in \mathbb{R} \end{align}
My attempt:
Using, trigonometric substitution
\begin{align}
\...
3
votes
2answers
101 views
Integration of $\cos x\cdot\cos 2x\cdot\cos 3x$
I study maths as a hobby and am trying to integrate
$\,\cos x\cdot \cos 2x\cdot \cos 3x\,$.
I am trying to split this up into a form in which I can integrate each component part. I have tried using ...
0
votes
1answer
108 views
Integral $\int_0^{\pi} \frac{\sin^2 x}{x^2} dx.$ [closed]
It is known that
$$\int_0^{\infty} \frac{\sin^2 x}{x^2} dx=\frac{\pi}{2}.$$
However, it is not so clear how fast $\int_0^{t} \frac{\sin^2 x}{x^2} dx$
converges to $\frac{\pi}{2}$ as $t$ goes to ...
1
vote
4answers
97 views
Solve $ \int\frac{1}{\sin(x)-\cos(2x)}dx $ …
Weierstrass substitution :
$$\tan(\frac{x}{2})=t$$ $$\sin(x)=(\frac{2t}{1+t^2})$$ $$\cos(x)=(\frac{1-t^2}{1+t^2})$$ $$dx=(\frac{2\,dt}{1+t^2})$$
Than : $$\cos(2x)=\cos^2(x)-\sin^2(x)$$
P.s I tried ...
6
votes
3answers
199 views
How to prove $\int _0^\infty \operatorname{si}(x) \operatorname{Ci}(x) \, dx=\ln 2$
How to prove the integral
$$\begin{align}&\int _0^\infty \operatorname{si}(x) \operatorname{Ci}(x) dx\\=&\int_0^\infty\left (\int_x^\infty\frac{\sin t}{t}dt \int_x^\infty\frac{\cos t}{t}dt\...
1
vote
2answers
47 views
How do I prove that $\int_{-\infty}^x \frac{\sin(ny)}{\pi y}\text{d}y$ tends towards the Heaviside step function?
I want some (ideally short) proof of the statement
$$\int_{-\infty}^x \frac{\sin(ny)}{\pi y}\text{d}y\,\,\stackrel{n\rightarrow\infty}{\longrightarrow}\,\,H(x)$$
with which I can later proof that $\...
6
votes
2answers
71 views
Evaluare $I(y)=\int_0^1\frac{1}{y+\cos(x)}dx$. Hence determine $J(y)=\int_0^1\frac{1}{(y+\cos(x))^2}dx$
Consider the following integral:
$$I(y)=\int_0^1\frac{1}{y+\cos(x)}dx $$
with Weierstrasse substitution I showed that
$$ I(y)=\frac{2}{\sqrt{y^2-1}}\arctan\left(\sqrt{\frac{{y-1}}{y+1}}\right). $$
It ...
4
votes
2answers
158 views
I can't find the pattern while evaluating $\int_0^{\pi/2}\sin^n(x)\,dx$ [closed]
While integrating:
$$
\int_0^{\frac{\pi}{2}}\sin^nx \, dx
$$
I have noticed that when $n$ is even the value is a multiple of $\pi$, but when $n$ is odd it is rational.
My results are:
n=1, 1
n=2, $\...
3
votes
1answer
68 views
Does $\int \tan^3x\sec^2x \space dx$ have 2 solutions?
So normally, you evaluate $\int\tan^3x\sec^2x \space dx$ by substituting $u = \tan x$ and $du = \sec^2x\space dx$ right? So,
$$\begin{equation}\begin{aligned}
\int\tan^3x\sec^2x \space dx &= \int ...
2
votes
0answers
96 views
Are there any tricks to integrate something like this?
How can we integrate?
Any tips would be helpful.
$$\int \frac{\cos(x) - \sin(x)}{\cos(x)+\sin(x)}\log(\sin(x)) \text dx$$
1
vote
3answers
79 views
How to prove $\int_{0}^{2\pi } \frac{ \sin(\theta - \phi) } { R^2 - 2rR \cos(\theta -\phi) +r^2 } d{\phi} =0$
This question was part of mock test of masters exam for which I am preparing and I am unable to solve it.
Show that $\int_{0}^{2\pi } \frac{ \sin(\theta - \phi) } { R^2 - 2rR \cos(\theta -\phi) +r^...
