Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [trigonometric-integrals]

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

5
votes
1answer
69 views

Evaluating $ \int \frac1{5S_{8} - 9S_{10} + 7S_{12} - 2S_{14}} {\rm d}x$, where $S_n = \sin^n(x) + \cos^n(x)$

Prove that $$ \int \frac1{5S_{8} - 9S_{10} + 7S_{12} - 2S_{14}} {\rm d}x = 2x - \arctan \left( \frac{\tan2x}{2 + \tan^22x} \right) + C$$ where $S_n = \sin^n(x) + \cos^n(x)$. Even ...
1
vote
2answers
51 views

Why $\int_0^{2\pi} \sin^2 x \mathrm{d}x \neq \int_0^{2\pi}\sin x \sin nx \mathrm{d}x \to n=1$

By orthogonality, we know that $$\int_0^{2\pi}\sin mx \sin nx \mathrm{d}x = \pi$$ iff $m=n$ and $0$ otherwise. Nevertheless, when I am calculating it as follows, I get $0$. $$\int_0^{2\pi} \sin x \...
6
votes
2answers
81 views

Integral of $\int\frac{\sin x}{\sin x+\cos x}dx$

The questions defines $$I=\int\frac{\sin x}{\sin x +\cos x}dx\;\;J=\int\frac{\cos x}{\sin x +\cos x}dx$$ It asked me to find $I+J$ and $J-I$ which I have done and I will show below but now I need to ...
2
votes
2answers
53 views

Integrating $\int \sec xdx$: Why is $\ln|\text{sec}x + \text{tan}x| + C$ preferred over $\tanh^{-1}(\sin x) + C$?

I was trying to integrate $\sec^3x$ and discovered that I would have to integrate $\sec x$ in the process. I had not seen the "standard" approach and came up with my own solution, which is apparently ...
0
votes
0answers
40 views

Integral $\cos\left(\ln\left(e^t+1\right)+at+b\right)dt$

I am trying to solve this integral: $$\int \cos\left(\ln\left(e^t+1\right)+at+b\right)dt $$ with $$a, b \in \Re$$ I can solve this: $$\int \cos\left(\ln\left(x\right)\right)dt $$ Defining: $u=\cos(...
-3
votes
0answers
26 views

indefinite integration involving atan

I need the indefinite integration of this: $$ \int \left[1 + \cos\left(K + \arctan\left(\frac{a+bx\ }{c + dx\ }\right)\right) \right] dx $$ If it helps, we can turn the form above into the following ...
1
vote
3answers
45 views

How do I integrate $\sqrt{1+\sec2x}$?

I tried converting $\sec 2x$ into $1/\cos 2x$, and then performing further simplification which led me to $$\int \sqrt{\frac2{1-\tan^2 x}} dx$$ I'm unable to take it from here.
4
votes
4answers
98 views

Prove that $\int_0^{\frac{n\pi}{4}} \frac{1}{3\sin^4(x) + 3\cos^4(x) -1} dx = \frac{n\pi}{4} $

Prove that $$\int_0^\frac{n\pi}{4} \left(\frac{1}{3\sin^4(x) + 3\cos^4(x) -1}\right) dx = \frac{n \pi}{4} $$ is true for all integers $n$. Here I've looked at the case where $n=1$ as $t=\tan(x)$ is ...
0
votes
1answer
34 views

A trigonometric function question for calculus [closed]

Compute the definite integral $$\int_0^1 \left(2-x^2\right)^{3/2}dx$$ Please help me, i suppose $x=\sqrt{2} \sin \theta$ but I just couldn't get the answer. Thank you
0
votes
1answer
30 views

Calculation of $\int_{-R}^R(R-x^2)^\frac{n-1}{2}dx$

Let $R>0$ be fixed, $n\ge 2$. What is $$\int_{-R}^R(R-x^2)^\frac{n-1}{2}dx?$$ I tried substitution $x=\sqrt{R}\sin(t)$. Then $dx=\sqrt{R}\cos(t)dt$ and $$\int_{-R}^R(R-x^2)^\frac{n-1}{2}dx=\int_{\...
2
votes
1answer
54 views

Calculate the integral of exp(z)/sin(z) over |z|=4 using the residue theorem.

Click to view the integral in correct format. Calculate the integral of exp(z)/sin(z) (as in the image above) over the positively oriented circle defined by |z|=4 using the residue theorem. This is ...
0
votes
0answers
30 views

About $ f(w,L) = \int_1^w \int_0^{2 L \pi} \frac{ \ln(\frac{\sin(x) +\ sin(vx)}{2} + \frac{5}{4})}{L(w - 1)} dx dv $

Consider $$ f(w,L) = \int_1^w \int_0^{2 L \pi} \frac{ \ln(\frac{sin(x) + sin(vx)}{2} + \frac{5}{4})}{L(w - 1)} dx dv $$ For real $w > 1 $ and integer $ L > 1$ Conjecture : $$ \lim_{L \to \...
0
votes
3answers
52 views

Can these integrals be solved by u-substitution using trig identities?

$\int \frac{\operatorname{sin} x}{2\sin x\cos x+5}dx$ $\int \frac{\operatorname{sin}x\cos x}{\operatorname{sin}x+\cos x}dx$ $\int \frac{dx}{\operatorname{sin} x+\cos x}$ I'd like to know if there's ...
1
vote
2answers
39 views

Contour intergal of a rational trigonometric function, can't find my mistake.

This is the integral : $$I = \int_{0}^{2\pi} \frac{dx}{(5+4\cos x)^2}\ $$ Which, according to wolfram alpha, should evaluate to $\frac{10\pi}{27} $, but the value i find is $\frac{20\pi}{27} $. These ...
0
votes
0answers
48 views

Trigonometric Integral Involving the fractional part

Let $\{\}$ denote the fractional part function, does the following Integral admit a closed-form ? $$\int_{0}^{\pi/2}\bigg\{\frac{1}{\cos(x)}\bigg\}dx$$
0
votes
3answers
54 views

Evaluate $I_n = \int_0^{\pi / 2} \sin n \theta \cos \theta \,d\theta$ by integrating by parts twice

By integrating by parts twice, show that $I_n$, as defined below for integers $n > 1$, has the value shown. $$I_n = \int_0^{\pi / 2} \sin n \theta \cos \theta \,d\theta = \frac{n-\sin(\frac{\pi ...
5
votes
1answer
69 views

If $I_{m,n} = \int_{0}^{\pi/2}\cos^mx\cos nx ~dx$, prove that $I_{m,n} = \frac{m(m-1)}{m^2-n^2}I_{m-2,n}$

If $$I_{m,n} = \int_{0}^{\pi/2}\cos^m x \cos nx ~dx$$ prove that $$I_{m,n} = \frac{m(m-1)}{m^2-n^2}I_{m-2,n}$$ I have tried my level best to solve but failed. Please help me out with a detailed ...
2
votes
1answer
49 views

Showing that $I_{n+2}+I_{n}=2I_{n+1}$, where $I_n=\int_0^\pi\frac{1-\cos n\theta}{1-\cos\theta}$ [closed]

I am unable to figure out to how to approach the problem, as all my attempts led to dead ends... Please help me out: Let $\displaystyle I_n =\int_0^\pi \frac{1-\cos n\theta}{1-\cos\theta} d\theta$ ...
12
votes
2answers
126 views

Showing $\int_0^{\int_0^u{\rm sech}vdv}\sec vdv\equiv u$ and $\int_0^{\int_0^u\sec vdv}{\rm sech} vdv\equiv u$

The two following very weird-looking theorems $$\int_0^{\int_0^u\operatorname{sech}\upsilon d\upsilon}\sec\upsilon d\upsilon \equiv u$$ $$\int_0^{\int_0^u\sec\upsilon d\upsilon}\operatorname{sech}\...
0
votes
0answers
44 views

Integration of $-\frac{32}{\pi^2 k^2} \int_k^{k\sqrt2} a\; L_2[a](\sqrt{{a^2}/{k^2}-1}+arcsin(\frac{k}{a})\;\;) da$

Hi I'm trying to solve this integral: $I=-\frac{32}{\pi^2 k^2} \int_k^{k\sqrt2} a \;L_2[a](\;\;\sqrt{\frac{a^2}{k^2}-1}+arcsin(\frac{k}{a})\;\;) da$ where k is a parameter constrained by $0<k<...
1
vote
0answers
49 views

$\int_0^x\sin^at\cos^bt\, \mathrm{d}t=?$

Consider the integral $$I(a,b)=\int_0^{\pi/2}\sin^at\ \cos^bt\ dt$$ As I have shown in numerous answers, $I$ has a close relationship with the beta function, namely $$I(a,b)=\frac12B\bigg(\frac{a+1}2,...
2
votes
4answers
73 views

Why does this method fail for finding the Fourier series for $\cos\left(\frac{x}{2}\right)$ on the interval $-\pi \lt x \lt \pi$?

This question is strongly related to this question (that does not have an answer). From "Riley, Hobson and Bence - Mathematical methods for physics and engineering", Section 12.5 - "Non-periodic ...
0
votes
4answers
65 views

Methods to solve $\int_{0}^{\infty} x^{n}\cos(x)\:dx$

I've been playing around with the following integral and was wondering if it can be generalised to any Real $n$. Does anyone know of any methods to approach this one? $$ I = \int_{0}^{\infty} x^n \...
3
votes
0answers
87 views

If a function is extended to make it periodic then must the integration limits also be extended?

The following extract is taken from "Riley, Hobson and Bence - Mathematical methods for physics and engineering", Section 12.5 - "Non-periodic functions", page 422 and 423: Find the Fourier series ...
1
vote
1answer
24 views

Simplifying Integration of $2xe^{3x}\sin(2x)\cos(2x)$

One of the problems on my homework was the following:Find $\int2xe^{3x}\sin(2x)\cos(2x)$. I am confused on how to approach this but I will leave my attempt in the answers section below. I am ...
0
votes
2answers
60 views

Find $\int_0^\pi \cos^4\theta \sin^3\theta~d\theta$ using de Moivre's theorem

Find $\displaystyle\int_0^\pi \cos^4\theta \sin^3\theta~d\theta$ using de Moivre's theorem. So I need to find and expression for $\cos^4\theta \sin^3\theta$ in terms of multiple angles. I know that $...
1
vote
0answers
25 views

$\int^{2\pi}_0\sin\frac{\theta}{2}\cdot\hat{r}d\theta$

I want to find: $$\int\limits^{2\pi}_0\sin\frac{\theta}{2}\cdot\hat{r}d\theta$$ while $\hat r$ is as usually used in Physics (i.e. for $\theta=0.5\pi$ you get $\hat r=\hat y$, and for $\theta=\pi$ you ...
2
votes
1answer
71 views

$\int\sqrt{1-\tan x}~\mathrm{d}{x}.$ (Integral of a trigonometric function under square root)

$\int\sqrt{1-\tan x}~\mathrm{d}{x}.$ is an integral which I am not able to solve. I have restricted my ideas on trigonometric substitution but cannot conclude to an answer...will really appreciate if ...
3
votes
2answers
101 views

How to integrate $\tan^{-1}\left(\frac{1}{2 \sin(x)}\right)$?

I want to calculate the following integral : $\displaystyle{\int^{\frac{\pi}{2}}_{0} \tan^{-1}\left(\frac{1}{2 \sin(x)}\right)} \text{ d}x$ But I don't how; I tried by subsituting $u = \frac{1}{2 \...
1
vote
3answers
60 views

How to calculate $\frac{I_{n+2}}{I_n}$ of $I_n = \int_{\frac {-\pi}{2}}^\frac{\pi}{2} cos^n \theta d\theta$

How do I calculate the $\frac{I_{n+2}}{I_n}$ of $I_n = \int_{\frac {-\pi}{2}}^\frac{\pi}{2} cos^n \theta d\theta$ ? [my attempt]: I could calculate that $nI_n = 2cos^{n-1}\theta sin\theta+2(n-1)\...
1
vote
5answers
64 views

Integrate $\int x\sec^2(x)\tan(x)\,dx$

$$\int x\sec^2(x)\tan(x)\,dx$$ I just want to know what trigonometric function I need to use. I'm trying to integrate by parts. My book says that the integral equals $${x\over2\cos^2(x)}-{\sin(x)\...
2
votes
1answer
44 views

On $s(\alpha)=\int_{0}^{\pi/2}\sin^{\alpha}(t)dt$

So, I am working on $$s(\alpha)=\int_{0}^{\pi/2}\sin^{\alpha}(t)dt$$ Looking for a general form. Although I am not (really) asking you to evaluate the integral, I have some questions about my methods. ...
0
votes
0answers
42 views

Area enclosed between the polar curves $r=3-3cos\theta$ and $r=4cos\theta$

Area enclosed between the polar curves $r=3-3cos\theta$ and $r=4cos\theta$ So the area of the two enclosed areas is the same so only one area needs to be calculated. the area of one loop then needs ...
0
votes
1answer
59 views

Integral of $\sin^5(x)\cos(x)$

I'm trying to solve the following integral: $$\int\sin^5(x)\cos(x)$$ I assumed I would do u-substitution where: $$u = \sin(x)$$ $$du = \cos(x) dx$$ Which would then cancel out the $\cos(x)$ And ...
8
votes
1answer
197 views

Another difficult 2D trigonometric integral

This is a follow-up question to A difficult 2d trigonometric integral. Unfortunately, I had a mistake in my calculations and I need a different (yet similar) seemingly simple integral solved: $$\...
1
vote
1answer
88 views

A difficult 2d trigonometric integral

I'm trying to solve the following seemingly simple integral - so far, without success: $$\int_{a}^{b}\int_{a}^{y}\frac{\cos(x-y)}{xy}\mathop{\mathrm{d}x}\mathop{\mathrm{d}y}$$ For some $0<a<b$....
1
vote
1answer
63 views

Evaluating $\int_{\pi/2}^{3\pi/2}\frac{1}{\pi\sqrt{1+d\cos(\gamma)}}d\gamma$

I am trying to evaluate the following integral. Any help will be highly appreciated. $$\int_{\pi/2}^{3\pi/2}\frac{1}{\pi\sqrt{1+d\cos(\gamma)}}d\gamma=?$$ where $d$ is any constant. My solution ...
0
votes
0answers
22 views

Integral with bilinear forms in exponent

I've encountered the following integral: $$ \int_{-\pi}^\pi d\omega \exp\{ \sum_{i,j=1}^n R_j H_{ij}^c(\omega) d_i + \sum_{i,j=1}^n I_j H_{ij}^s(\omega) d_i \},$$ where $R_j, I_j, d_i$ are reals and $...
5
votes
2answers
120 views

Solving the Integral without Cauchy Integral formula.

I saw this integral in a competitive exam which I donot recall, but I found this problem in my notebook which has to be shown without using Cauchy integral theorem, Poisson integral formula over ...
3
votes
1answer
92 views

Evaluating $\int_{-\infty}^\infty e^{i(ax^2+bx+c)}\frac{\operatorname{sin}^2x}{x^2}dx$

Is there a closed form for the following integral? $$\int_{-\infty}^\infty e^{i(ax^2+bx+c)}\frac{\operatorname{sin}^2x}{x^2}dx$$ Nothing of this form seems to appear in Gradshteyn and Ryzhik.
5
votes
2answers
66 views

How to integrate $\int_{1}^{e} (x+1)e^{x}\ln{x}dx $?

How to integrate $$ \int_{1}^{e} (x+1)e^{x}\ln{x}dx$$ I used following ways: integration by parts I first split the function into $$ \int_{1}^{e} (x)e^{x}\ln{x}dx + \int_{1}^{e} e^{x}\ln{x}dx$$ ...
3
votes
2answers
58 views

Finding $\int \sin^5(2x)\cos^3(3x)~dx$

In solving the integral $$\int\sin^5(2x)\cos^3(2x)~dx,$$ I misread it as $$\int\sin^5(2x)\cos^3(\color{red}{3x})~dx.$$ After a while of failing to solve this, I realized my mistake. How would I ...
0
votes
1answer
21 views

Trigonometric Substiution

I am currently working on a practice problem on trig substitution. I have no problem solving it my way but I don’t see how i can use the below stated identity and partial integration. (I am sorry i ...
1
vote
2answers
94 views

Contour Integral of irrational polynomial from -1 to 1

I've been stuck at htis contour integral problem for a few hours now, and seem to be hitting brick walls. $$ \int_{-1}^1 \frac{\sqrt{1-x^2}}{1+x^4}dx\,, $$ I tried a trig substitution $x=\cos{\theta}...
7
votes
3answers
86 views

How to integrate $ \int \frac{\sin {3x} + \cos{3x}}{ \sin^3 x + \cos^3 x } dx\ $

I want to know how to integrate this function I have tried many things many substitutions but none works. I even try to expand the numerator by $\sin3x$ and $\cos3x$ properties and tried to convert ...
-3
votes
1answer
215 views

A different method of evaluating $ \int\frac{1}{\sqrt{\sin^3(x)\cdot\sin(x+\alpha})}\,dx$

I was practicing indefinite integrals and came to this integral: $$ \int\frac {1}{\sqrt{\sin^3(x)\cdot\sin(x+\alpha})}\,dx. $$ Everywhere there is only one method given for solving this ...
3
votes
6answers
95 views

$I_n = \int_{0}^{\frac{\pi}{2}}(\cos t)^n \ dt$ converges to 0?

How one can prove that the sequence $\left ( I_n \right )$ defined as $$ I_n = \int_{0}^{\frac{\pi}{2}}(\cos t)^n \ dt, $$ $n \in \{ 0,1,2,...\}$ converges to $0$? Is easy to show, by the way, that ...
-1
votes
3answers
158 views

How to evaluate $\int\frac{ \sin^8 x - \cos^8x } { 1 - 2\sin^2 x \cdot \cos^2 x }\mathrm{d}x$? [closed]

I would like to calculate the following integral: $$ \int\frac{ \sin^8 x - \cos^8x } { 1 - 2\sin^2 x \cdot \cos^2 x }\mathrm{d}x. $$ I have tried to rewrite the integral as: $$ \int\...
8
votes
2answers
111 views

How to calculate the integral $\int \frac{a\tan^2{x}+b}{a^2\tan^2{x}+b^2} dx$?

How to calculate the integral $\displaystyle\int \dfrac{a\tan^2{x}+b}{a^2\tan^2{x}+b^2} dx$? It seems like a $\arctan$ of what else... but I can not work it out. I have previously asked how to ...
1
vote
0answers
99 views

Why does $\frac{\sin x}{x}$ have no anti-derivative [duplicate]

I've been messing around with indefinite integrals. Watching some youtube videos and I found the Sinc function and that it has no finite Anti-derivative. Desmos being my favourite program ever I ...