3
votes
1answer
55 views

Nonsensical result in the midst of calculating an integral via substitution.

I was just calculating an integral via a trigonometric substitution and ended up with $\color{red}{ \text{something pretty nonsensical} }$ but $\color{blue}{ \text{reversing the substitution} }$ ...
3
votes
6answers
137 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 ...
2
votes
0answers
97 views

Evaluating $\int_0^x \lvert \cos t \rvert dt$

in my mathbook there is given a solution to $$\int_0^x \lvert \cos t \rvert \, dt $$ but without any hints or tips. $$\int_0^x \lvert \cos t \rvert \, dt = \sin\left(x - \pi \left\lfloor \frac x ...
7
votes
3answers
67 views

How to find the integral $\int_0^{70 \pi} |\cos^{2}x\sin x|\,dx$?

I need help with this problem: $$\int_0^{70 \pi} \left|\cos^{2}\!\left(x\right)\sin\!\left(x\right)\right| dx$$ My friend says it's 140/3 but I don't see how.
1
vote
3answers
84 views

Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let $$ \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} $$ I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
3
votes
1answer
51 views

Why is the value of $\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|\,dx$ independent of integer parameter $n$?

I am not able to find an easy solution for the following formula $$\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|dx=4+\frac{4}{3}\pi\sqrt{3}.$$ Please help me prove it. Why it does not depend on the (positive) ...
6
votes
2answers
106 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 ...
15
votes
0answers
217 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} ...
3
votes
3answers
51 views

The average value of the function $y=\tan(2x)$ over the interval $[0,\frac{\pi}{8}]$

I was given the following question in a technology free exam. How would one go about solving this without the use of a calculator? NB. I am currently in my last year of high school so please don't ...
8
votes
2answers
169 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
1answer
80 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 ...
7
votes
2answers
208 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 ...
15
votes
4answers
587 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
101 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\\ ...
1
vote
2answers
72 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 ...
4
votes
2answers
89 views

Evaluate $\int_{0}^{\pi}\sin^5{\theta}\cos^2{\theta}\ d\theta$ [duplicate]

I'm trying to find the mass of a spherical object with a given density function, and to do so I must solve this integral $$\int_{0}^{\pi}\sin^5{\theta}\cos^2{\theta}\ d\theta,$$ but no matter which ...
2
votes
1answer
30 views

Double integral compute

I'm struggling with this one for a week: There is a range $R$ that it's points $(x,y)$ are defined as: For each $0 \le x \le 32$, all the values of $y$ are $\sqrt[5]{x} \le y \le 2$. We need to ...
3
votes
1answer
72 views

Closed form of a trigonometric integral sought

I am trying to evaluate the definite integral $I(a,b)$, with $a,b\in\mathbb{R}$, defined by $$I(a,b):=\int_{0}^{2\pi}\sqrt{1-(a+b\cos{\theta})^2}\mathrm{d}\theta.$$ Assume $a,b$ are suitably ...
7
votes
2answers
209 views

Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$

$$\int_0^1 \frac{\arcsin(x)}{x}dx$$ This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma ...
0
votes
1answer
46 views

Definite integral

So I was playing around with Euler's Reflection Formula ($\Gamma(s)\Gamma(1-s)=\frac{\pi}{\sin(\pi s)}$), trying to prove it with calculus, and was able to reduce $$ ...
2
votes
3answers
74 views

How do I evaluate integrals that involve the signum ($\text{sgn}$) function?

For example, I want to evaluate $$ \displaystyle \int_{0}^{2\pi} \left| \sin x \right| \text{ d}x $$ and I already know that: $$ \displaystyle \begin{aligned} \int \left| \sin x \right| \text{ d}x ...
4
votes
0answers
77 views

A trigonometric integral with sin(cos(x)) in exponent

Evaluate: $$\int_0^{\pi} x\csc^{\sin(\cos x)}(x)\,dx$$ I honestly don't know how to deal with this case. If I apply the property $\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$, I get: $$\int_0^{\pi} ...
4
votes
4answers
90 views

Convolution integral $\int_0^t \cos(t-s)\sin(s)\ ds$

How can I calculate the following integral? $$\int_0^t \cos(t-s)\sin(s)\ ds$$ I can't get the integral by any substitutions, maybe it is easy but I can't get it.
7
votes
3answers
95 views

Integration of $\int_{0}^{\frac{1}{2}}\frac{\sin^{-1}(x)}{\sqrt{1-x^2}} dx$ ??

I was solving the integration of inverse trigonometric function and faced a question which i find it hard to understand. I need to find the definite integration of this function. ...
1
vote
2answers
50 views

Definite Trig Integrals: Changing Limits of Integration

$$\int_0^{\pi/4} \sec^4 \theta \tan^4 \theta\; d\theta$$ I used the substitution: let $u = \tan \theta$ ... then $du = \sec^2 \theta \; d\theta$. I know that now I have to change the limits of ...
4
votes
4answers
80 views

Trig integral with sine and cosine

What sort of formulas can I use to reduce this into something I can work with? $$3a^2\int_{0}^{2\pi} \sin^2(\theta)\cos^4(\theta) \, d\theta$$
1
vote
3answers
91 views

Evaluation of an integral $\int \sin^2(x) \sqrt{1+\alpha^2 \cos^2(x)} \mathrm d x$

I am currently trying to get a general expression of the following integral, I spaned many questions with the above tags and found nothing close to it: $$ I_n = \int_0^{n \pi} \sin^2(x) ...
14
votes
2answers
515 views

A tricky integral

I'm trying to find the exact value of $$\int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{\arctan{(x^2)} }{1+x^2} \, dx$$ Ostensibly, I'd want to use this: $$\frac{d}{dx}\arctan{(x)}=\frac{1}{1+x^2}$$ But ...
-1
votes
2answers
57 views

Why is the integral of $\cos(x) - \sin(x)$ from $0$ to $π/4$ equal to $\sqrt2 - 1$?

Why is $$\int_ {0}^{π/4} {\cos(x)} - {\sin(x)} \ \mathrm{d}x=\sqrt2 -1$$ This answer popped up on a problem I was doing and it piqued my interest. Can anyone help me out?
7
votes
3answers
113 views

Why is $\displaystyle\int^{\infty}_{0}{(1-\cos x)\over{x^{2}}}dx = \frac\pi{2}$?

I have been having trouble understanding Fourier series and analysis in one of my classes. This is one problem from the text and we have to show that this is true. I have done other problems related ...
0
votes
1answer
44 views

Question regarding trigonometry

I've got this thing on my mind : we know that $cos(x)$ is a periodic function , hence integral from $2(k-1) \pi$ to $2k \pi$ will yield the same value for any $k \geq1$. My question is , why is ...
0
votes
4answers
144 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
41 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
73 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
104 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
146 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
21 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
38 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
84 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
46 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
102 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
211 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
884 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
63 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
60 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
85 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
46 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
245 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
109 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( ...