6
votes
5answers
122 views

How to find $\int|\cos x|\,dx$?

How do I find closed form for $\int|\cos x|\,dx$ for all real $x$? It can be expressed as incomplete elliptic integral of the second kind: $$\int|\cos x|\,dx=\int\sqrt{1-1^2\sin^2x}\,dx=E(x,1)$$ ...
1
vote
0answers
49 views

Find $\int \tan(\tan x)\hspace{1mm}dx$

Find $\int \tan(\tan x)\hspace{1mm}dx$ This is an Interesting problem, which I have been trying from different directions, nothing seems to work, its been a day on this one. Can anyone figure out ...
0
votes
1answer
27 views

How to prove that this equality is the development of a fourier series?

how can I show that this identity is a development of a fourier series? $$f(x)=\sin^3 x=\frac{3}4 \sin x-\frac{1}4 \sin 3x$$ I tried this: obtain the Fourier coefficients whih $$b_n=\frac{2}\pi ...
2
votes
1answer
49 views

Real life math to explore/solve

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
3
votes
1answer
61 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
4
votes
3answers
87 views

Find $\int \sinh^{-1}x\hspace{1mm}dx$

Find $\int \sinh^{-1}x\hspace{1mm}dx$ $ $ I am asked to use the following Equation: $$\int \tan^{-1}x\hspace{1mm}dx= x\tan^{-1}x-\ln(\sec(\tan^{-1}x))+C$$ $ $ The confusing part is : What has ...
0
votes
0answers
51 views

Evaluate $\int\tan x\, dx$

$-1 = 0$ by integration by parts of $\tan(x)$ The solutions to the above problem state that you can't cancel the integrals on each side because they both have an unknown constant attached to them. ...
1
vote
3answers
43 views

How to integrate $\int\frac{t^2}{\sqrt{4t-t^2}} \, dt$ using trig substitution.

How do I integrate $\int\frac{t^2}{\sqrt{4t-t^2}} \,dt$? I solved this integral by a very long process(over 2 pages of work) and I got the answer of ...
1
vote
2answers
53 views

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal?

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal? I know both of these are the answer to $\int \csc x \space dx$, and I am able to work them out with calculus ...
1
vote
3answers
53 views

Evaluating $\int \:\sqrt{1+e^x}dx$ , why I got different answers?

I've got 2 steps to evaluating $\int \:\sqrt{1+e^x}dx$ which lead to different values first step : $\int \:\sqrt{1+e^x}dx$ let $u\:=\:\sqrt{1+e^x}$ , $du\:=\:\frac{e^x}{2\sqrt{1+e^x}}dx$ , but ...
2
votes
1answer
46 views

Integrating $\int\frac{\sqrt{16x^2-9}}{x}dx$?

I am trying to differentiate from my previous question, but I am having trouble in the finishing steps. I have the integral $\int\frac{\sqrt{16x^2-9}}{x}dx$. $$v=4x \hspace{15pt}dv=4dx$$ ...
2
votes
3answers
66 views

$\int \dfrac{\cos x}{\left(\cos (2x)\right)^{3/2}} dx$

Wolfram gives this nice result: $$\int\frac{\cos x dx}{\cos^{3/2}2x}=\frac{\sin x}{\sqrt{\cos 2x}}+\text{constant}$$ I have tried writing $\cos 2x = \cos^2x - \sin^2x $ and doing Weierstrass ...
0
votes
1answer
24 views

Integrate a trigonometrical expression [closed]

I know that this may sound a silly question but is the following which is the integral: $\int(1+(\cot^2)x)dx$
2
votes
3answers
42 views

Evaluating $\int \:x\csc \left(x^2\right)\cot \left(x^2\right)dx$

I try to evaluating $\int \:x\csc \left(x^2\right)\cot \left(x^2\right)dx$ let $u=x^2,\quad \quad du=2xdx,\:\quad \:dx=\frac{1}{2x}du$ then i get $\int \:x\csc \left(u\right)\cot ...
2
votes
3answers
53 views

Calculate wine volume in a horizontal barrel using a dipstick

I suck at math, but still need a way to mark a dipstick to measure the volume of wine in a barrel. This question has been asked, but the only answer is to cryptic for me to understand! My barrel has ...
3
votes
3answers
276 views

Evaluating $\int \frac{1}{x\sqrt{9x^2-1}}\,dx$

I try to integrate $$\int \frac{1}{x\sqrt{9x^2-1}}\,dx$$ let $u=x^2,\quad \quad du=2x\,dx,\:\quad \:dx=\frac{1}{2x}\,du$ $$ \begin{align} & \int \frac{1}{x\sqrt{9u-1}}\frac{1}{2x}\,du \\[8pt] = ...
-1
votes
3answers
72 views

How to evaluate an integral of the form $\int \frac{dx}{-ax^2 + b}$?

I need to evaluate $\int \frac{dx}{-ax^2 + b}$ while both $a$ and $b$ are positive. I was blocked while I was trying $ x=\tan\theta $ which turned $ dx=\sec^2\theta\, d\theta $ This method didn't ...
4
votes
2answers
92 views

Trigonometric integral: $\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$

Is it possible to evaluate the following in a closed form? $$\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$$ I found the above definite integral at I&S but the solution is ...
3
votes
2answers
51 views

How to simplify the integral of $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$?

So I am trying to integrate this problem $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$, and my professor went over it in class and went from $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx \rightarrow ...
3
votes
3answers
95 views

Integral of $\cos(\cos x)$ over $[0,2\pi]$

How to compute the following integral? $$\mathcal{J}_2=\int_{0}^{2\pi}\cos(\cos t)\,dt$$ I'm trying to compute this integral, but I have no idea of how to do it, can someone help me?
3
votes
5answers
162 views

Evaluate$ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$

Find the value of the integral $ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) $ I tried putting $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but am unable to proceed further. I think the following integral can be ...
6
votes
4answers
93 views

Integral of Sinc Function Squared Over The Real Line

I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$ Would a contour work? I have tried using a contour but had no success. Thanks. Edit: About 5 minutes after posting this ...
5
votes
4answers
105 views

Evaluate $\int{\sin^3(x)\cos^2(x)}dx$

I'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!
2
votes
2answers
83 views

Integration of $1/\sin^3 x$

I need a explanation of this problem: $$ \int \frac{1}{\sin^3 x}\,dx $$ Change the variable $$ t = \tan (x/2) $$ With use of $\tan$, $\cos$, $\sin$ and $\cot$, only. So how do I ...
3
votes
0answers
82 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
0
votes
2answers
57 views

The limit of Riemann sums $\sum_{k=1}^{n}\cos(\frac{k\pi}{2n})\frac{ \pi}{2n}$

Find the limit of Riemann sums $$\lim_{n\rightarrow \infty} \sum_{k=1}^{n}\cos(\frac{k\pi}{2n})\frac{ \pi}{2n}$$ on the interval $$[0,\frac{\pi}{2}]$$ Progress All I have managed to do is ...
6
votes
2answers
118 views

Having fun integral $\int_0^{\pi/4} \cos x \arctan(\cos x)\, dx$

Playing around with the inverse trigonometric function integration, I found a nice closed-form of the following integral $$\int_0^{\pi/4} \cos x \arctan(\cos x)\, ...
4
votes
6answers
146 views

How do I solve $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{4\,dx}{\sin^2(x)\cos^2(x)}$?

Alright so I have $$\int_{\pi/6}^{\pi/4}\frac{4\,dx}{\sin^2(x)\cos^2(x)}.$$ And I am not completely sure on how to tackle this problem. All I have done thus far is ...
2
votes
3answers
86 views

Trigonometric integral evaluation: $\int 4 \sin^4 x \cos^3 x \,dx$ [duplicate]

Evaluate the following integral $$\int 4 \sin^4 x \cos^3 x \,dx$$ I can do simple integration problems, but problems like this seem to stump me, I created this problem so I could solve and compare it ...
2
votes
1answer
80 views

Evaluate $\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx $

Evaluate the indefinite integral $$\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx$$ I tried multiplying and dividing by $\sec^2 {x} $ and then setting $\tan{x}=y$ but no good. Then I set $\cos ...
19
votes
0answers
242 views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
2
votes
3answers
191 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
8
votes
2answers
200 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
0
votes
0answers
58 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
2
votes
2answers
64 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
0
votes
1answer
83 views

Finite integral with goniometric functions, $\int_0^{\infty} \frac{8\sin^4(\pi f t)\tan^2(\pi f/2)}{(\pi^4 \tau^2 f^3) }df$

I have difficulties trying to find an algebraic solutions of the following integral: The $\tau$ in this formula is an integer, which is a very important fact because only then this integral is ...
2
votes
4answers
111 views

Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
3
votes
3answers
79 views

What is the value of $ \int_{x}^{1} \arcsin \left( \frac{2t}{t^2+1} \right) \text{d}t $?

Is this result true? Wolfram doesn't seem to be able to evaluate the definite integral in the allowed time. $$ \int_{x}^{1} \arcsin \left( \dfrac{2t}{t^2+1} \right) \text{d}t = \dfrac{\pi}{2} - ...
7
votes
2answers
213 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
3
votes
2answers
63 views

Evaluate integral by completing the square and doing trigonometric substitution

$\int \frac{1}{(x-2)\sqrt{x^{2}-4x+3}} dx$ is my problem Complete the square $\int \frac{1}{(x-2)\sqrt{(x-2)^{2}-1}} dx$ I know I'm probably supposed to use $ \frac{d}{dx}\operatorname{arcsec}(u) = ...
0
votes
2answers
77 views

Integrate $\int \sin^4x \cos^2x dx$

Integrate $$\int \sin^4x \cos^2x dx$$ Now, there's few solutions to this problem already on the internet. For example on yahoo: https://answers.yahoo.com/question/index?qid=20090204203206AAbjUfM and ...
1
vote
3answers
119 views

Evaluate $\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$

$$\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$$ My approaches so far has been using substitution with $\tan x = t$ and $\tan \frac x2 = t$ but the calculations has been harder than I ...
1
vote
1answer
58 views

Rewriting a double integral with complex exponential function

Why can we write $$ \begin{align} I_T &= \int_\mathbb{R}\int_{-T}^{T}\frac{e^{-ita}-e^{-itb}}{it}e^{itx}dtdF(x)\\ &= \int_\mathbb{R}\left[\int_{-T}^{T}\frac{\sin(t(x-a))}{t}dt - ...
17
votes
4answers
649 views

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
6
votes
1answer
110 views

Evaluate: $I = \int^{\pi/2}_0 (\sqrt{\sin x}+\sqrt{\cos x})^{-4}dx$

Evaluate : $$I = \int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx$$ Attempt : \begin{align} I&=\int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx\\ ...
4
votes
2answers
77 views

Integration practice of $\int \frac{\sqrt{25-y^2}}{y}dy$

I need to solve $\int \frac{\sqrt{25-y^2}}{y}dy$. I originally thought IBP, but that led to a very large and confusing algebra problem. Then I started to look at the $\sqrt{25-y^2}$ and started to ...
1
vote
2answers
76 views

if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $?

I would be interest to show : if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $ ? my second question that's make me a problem is that : what is :$ f^{-1}(\pi) $ ? I would be ...
2
votes
2answers
118 views

Mathematical Identity

I'm stuck in a path on a paper about thermal conductivity. There is a identity involving an integral that a I can't realize how they've perfomed it. Here is it: $$\lim_{N\to \infty} ...
1
vote
2answers
36 views

Substitution of an implicit variable

I wasn't sure how to title this question: I want to manipulate the integral $$I(a,b) = \int_0^{\frac{\pi}{2}} \frac{d \phi}{\sqrt{a^2\cos^2 \phi + b^2 \sin^2 \phi}}$$ with this subsitution: $$\sin ...
2
votes
4answers
97 views

integration by parts of trig functions

Can anyone help me with this integral? $\int{x^3 \sin(x^4) dx}$ I set $u=x^3$, and I let $v=-\cos(x^4)$, so that $\frac{dv}{dx}=\sin(x^4)$ I tried using integration by parts, but, whenever I come ...