1
vote
2answers
66 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
115 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
34 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
83 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 ...
1
vote
2answers
29 views

Line integral of a vector function involving sine and cosine

I have line integral of a vector function: $\vec{F}=-e^{-x}\sin y\,\,\vec{i}+e^{-x}\cos y\,\,\vec{j}$ The path is a square on the $xy$ plane with vertices at $(0,0),(1,0),(1,1),(0,1)$ Of course it is ...
0
votes
2answers
81 views

How to integrate this using u substitution? [closed]

How can I solve this using u substitution or trigonometry? $$\int e^{2x} \cos(e^{2x}) dx = \frac 1 2 \sin(e^{2x}) + \text{constant}$$
4
votes
2answers
80 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
26 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
69 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 ...
3
votes
3answers
45 views

Evaluate the integrals $\int \sin{x} \cot^2{x} \,dx$ and $\int \cos{x} \cot^2{x} \,dx$.

Can you please show how to evaluate the integrals $$\int \sin{x} \cot^2{x} \,dx$$ and $$\int \cos{x} \cot^2{x} \,dx.$$
2
votes
3answers
108 views

Is this integral with sine and cosine such a challenge?

...or maybe I just don't know some specific trick with trigonometric functions? Well, anyway, here it is: $$\int{\sin^6{x}\cos^4{x}\, dx}$$ I'm bored with it, because I get 9 integrals out of 1 ...
7
votes
2answers
192 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
45 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 $$ ...
7
votes
1answer
103 views

Integrate : $\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}dx$

$$\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}\ dx$$ My approach : Dividing the denominator by $\cos^2x$ we get $\dfrac{x^2\sec^2x }{(x -\tan x)(x\tan x +1)}$ then $$\int ...
1
vote
0answers
54 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
3
votes
2answers
220 views

How to integrate this formula with secant, exponential, and tangent?

How to integrate this? $$\int \sec^2(3x)\ e^{\large\tan (3x)}\ dx$$
1
vote
3answers
61 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
65 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
78 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.
1
vote
0answers
20 views

A Hard integral in 2-D.

I'm having a trouble integrating (in $\mathbb{R}^2$) the following formula: $$\frac{t}{|B(x,t)|}\int_{B(x,t)} \frac{||y||}{(t-||x-y||^2)^{\frac{1}{2}}} dy $$ where $B(x,t)$ is the ball with center ...
0
votes
3answers
69 views

Trig Substitution Integration

Didn't find this one here so I'm asking away: I tried integrating by substitution by ended up with just $x + C$ which is clearly wrong. My work, as requested:
7
votes
3answers
89 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
44 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 ...
0
votes
0answers
101 views

Integrating quotients with polynomials in numerator and denominator that are raised to fractional powers

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure ...
4
votes
4answers
78 views

Trig integral with sine and cosine

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

Is there a way besides integration by parts to solve this integral?

$$\int_{0}^{2\pi} -10\cos^9(t)\sin^4(t)t^4\,dt$$ Maybe a formula for this form or something?
4
votes
1answer
54 views

Trigonometric substitution for integral question.

I'm reviewing my quizzes to study for midterm tomorrow, and I came across a problem where I'm supposed to integrate: $$\int\frac{1}{x^2\sqrt{4-x^2}}dx$$ I used Mathematica to solve the problem and ...
3
votes
2answers
64 views

Trigonometric integral with the function- sin

I need some help with this integral please: $\displaystyle\int x\sin\frac{1}{x}dx$
2
votes
1answer
96 views

Integrating $\sqrt{1+\cos^2}$

As part of my Calculus II final we had a bonus question that had $$\int \sin(x)\sqrt{1+\cos^2(x)} \, dx \tag{*} $$ This set-up integral was not given and I know that (*) is easy to solve with ...
2
votes
0answers
19 views

integrate the square of angular distance from the node of a spherical triangle

Guessab, Noouisser, and Schmeisser "A Definiteness Theory for Cubature Formulae of Order Two", Constructive Approximation (2006)24:263-288 Define a quantity $R[||\cdot||^2]$ which is $$\sum_{i=1}^N ...
1
vote
3answers
79 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
500 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
56 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?
8
votes
3answers
502 views

Odd $\sin/\cos$ integral

How to evaluate $$\int \frac{\sin^3 x}{\cos^5x}dx\ ?$$ I've tried various substitutions with $\sin x = u$ or $\cos x = u$, I've tried using Euler's formula which result in too heavy calculations and ...
1
vote
3answers
92 views

Integral with goniometric functions $\int(1+\cos^2x-\sin^2x)dx$

I am solving this example: Transcription: \begin{align} &\int(1+\cos^2x-\sin^2x)dx=\int(1+1-\sin^2x-\sin^2x)dx=\int(2-2\sin^2x)dx=\\ ...
5
votes
5answers
183 views

$\int\frac{1}{\sin(x-a)\sin(x-b)}\,dx$

I'm stuck in solving the integral of $\dfrac{1}{\sin(x-a)\sin(x-b)}$. I "developed" the sin at denominator and then I divided it by $\cos^2x$ obtaining ...
-1
votes
4answers
58 views

Homework help, Trigonometric intergral [closed]

Can someone please help me with this question. $$ \int \ \frac{16}{1-\cos8x} \ \ dx \ \ . $$ Thank You
1
vote
2answers
85 views

Reduction formula tricky problem (Further Maths: F3)

$\int \:e^{ax}\cos ^n\left(x\right)dx$ I just cannot get it to reduce, I keep ending up with too many species in the next integral to use parts again. I have important Further Pure F3 exam in a month ...
1
vote
2answers
46 views

Integration with trigonomitry

Hi I was trying to find the area between the following curves (below) however I am unsure how to continue from the trigonometry which gets presented: Curves: $$y = 2\sin(x)\\y = ...
2
votes
4answers
115 views

How to find the integral for $\int 2^{\sin{x}}\cos{x}\;\mathrm{d}x$?

What would be the ideal approach in finding the integral for: $$ \int 2^{\sin{x}}\cos{x}\;\mathrm{d}x $$
7
votes
3answers
111 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 ...
1
vote
1answer
33 views

Integrating an equation with both cos and tan

$$\int2\cos^5x\cdot\tan^6x\cdot dx$$ $$2\int\cos^5x\cdot\frac{\sin^6x}{\cos^6x}\cdot dx$$ $$2\int \frac{\sin^6x}{\cos{x}} dx$$ $$2\int\cos^{-2}x\cdot \sin^6x\cdot \cos{x}\cdot dx$$ ...
7
votes
2answers
225 views

Integral $ \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}$

$$ I\equiv \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}. $$ This integral does have a closed form. I am not sure where to start. We can factorize the denominator as $$ \cos^3 x+2\sin(2x)-5\cos ...
0
votes
3answers
140 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
2
votes
2answers
58 views

Why are differential of $\sin^2(x)$ and integral of $\sin(2x)$ not the same?

I was working on a list of common integrals and differentials and I came across this question. If $${d\over d\theta}(\sin^2\theta) = \sin(2\theta)$$ Then why is $$\int \sin(2\theta) \space d\theta = ...
1
vote
2answers
40 views

Is there a way to integrate $\cos^{2} {3x}$ with a different technique than integration by parts?

The question is just as it is on the title: Is there a way to integrate $\cos^{2} {3x}$ with a different technique than integration by parts? And in case there is, how can I do it?
4
votes
2answers
43 views

check my solution to indefinite integral problem with arccos

So we had homework it asked us to find $$\int\arccos(x)dx$$ I have found that $$\int\arccos(x)dx=x\arccos (x)+\sqrt{1-x^2}+c$$ Is this right?
1
vote
2answers
54 views

How to evaluate this integral using trigonometric substitution?

I am pretty sure that my answer is correct but given answer for the exercise from textbook Calculus James Steward was slightly different. Any idea to solve this: $$\int\frac{x}{\sqrt{x^2+x+1}} \, ...
1
vote
1answer
44 views

Problem with solving a complicated Integral

I need to determine the $ \int \frac{\sin^3(x)}{8-\cos^3(x)} dx$. It's an indefinite integral.
2
votes
2answers
55 views

Integration Problem with a Trig substitution

Okay I am a little stuck on this problem. $$\int \tan^5(x)\sqrt{\sec(x)} \; dx$$ What should be my first step for a u sub or a trig sub? I have tried to use $u=\sec(x)$ and then $u=\tan(x)$, but I ...