Questions tagged [trigonometric-integrals]

Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.

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3
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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
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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
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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
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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 ...
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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\...
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2answers
32 views

Evaluate the integral using Trigonometric Substitution [closed]

$$\int_{0}^{3}\frac{x^3}{(3+x^2)^{\frac{5}{2}}}dx$$
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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*}
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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
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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
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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
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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
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1answer
74 views

Find the value: $\int_{0}^{π/6}\ 4\sin^{2}xdx$

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!
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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 ...
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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
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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\...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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1answer
41 views

Euler’s gamma functions and exponential integrals

I am trying to solve these integrals, where $ (C_i,S_i) ≡ (\cos λp_i, \sin λp_i)$ I have done $D_{SS}$. But in $D_{SC}$, I encounter some divergences. The author says this in the paper: The ...
2
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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
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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
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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
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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
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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
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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
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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
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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
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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^...

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