Tagged Questions
0
votes
3answers
65 views
How to solve these?
Inverse Trigonometric Functions
They are incomplete and I don't know how to complete them.
Who can help me?
1st
$$
\int\frac 1{ x \sqrt{x^{6} - 4}}dx
$$
I tried with:
$$u = x^3 $$
$$du= 3x^2dx$$
...
3
votes
3answers
56 views
Integrate $\int {{{\left( {\cot x - \tan x} \right)}^2}dx} $
$\eqalign{
& \int {{{\left( {\cot x - \tan x} \right)}^2}dx} \cr
& = {\int {\left( {{{\cos x} \over {\sin x}} - {{\sin x} \over {\cos x}}} \right)} ^2}dx \cr
& = {\int {\left( ...
5
votes
4answers
72 views
Integrate ${\sec 4x}$
How do I go about doing this? I try doing it by parts, but it seems to work out wrong:
$\eqalign{
& \int {\sec 4xdx} \cr
& u = \sec 4x \cr
& {{du} \over {dx}} = 4\sec 4x\tan 4x ...
2
votes
3answers
40 views
Integrating a sine function that is to an odd power
I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.
$\eqalign{
& ...
3
votes
3answers
102 views
How do you integrate the following trigonometric function involving sin and cos?
How do you integrate the following functions:
$$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^2 \, d\theta$$ and $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^3 \, d\theta
$$
...
5
votes
1answer
75 views
Find the following integral: [duplicate]
Find $$\int \sqrt{\tan x}dx$$
My attempt:
$$\text{Let}\ I=\int \sqrt{\tan(x)}dx$$
$$\text{Let}\ u=\tan(x), du=(1+\tan^{2}(x))dx$$
$$I=\int \frac{\sqrt{u}}{u^{2}+1}$$
$$\text{Let}\ v=\sqrt{u}, ...
2
votes
1answer
40 views
Trigonometric Integration + Series
I am doing an integration question:
$$\int \frac{1-\cos^{2m}x}{1-\cos^2x}$$
So I have to show that $$\frac{1-\cos^{2m}x}{1-\cos^2x}=1+\cos^2x+\cos^4x+...+\cos^{2(m-1)}$$
How can I do that?
7
votes
6answers
145 views
Find the following integral: $\int {{{1 + \sin x} \over {\cos x}}dx} $
My attempt:
$\int {{{1 + \sin x} \over {\cos x}}dx} $,
given : $u = \sin x$
I use the general rule:
$\eqalign{
& \int {f(x)dx = \int {f\left[ {g(u)} \right]{{dx} \over {du}}du} } \cr
...
13
votes
3answers
218 views
Proving a trig infinite sum using integration
How can I prove the following using integration and elementary functions?
Prove that:
$$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$
$0 < \theta < 2\pi$
2
votes
3answers
95 views
Fractional Trigonometric Integrands
$$∫\frac{a\sin x+b\cos x+c}{d\sin x+e\cos x+f}dx$$
$$∫\frac{a\sin x+b\cos x}{c\sin x+d\cos x}dx$$
$$∫\frac{dx}{a\sin x+\cos x}$$
What are the relations between the numerator in the denominator, and ...
0
votes
2answers
80 views
Trapezoid rule over trigonometric polynomials
The question is regarding trapezoid rule applied on trigonometric polynomials
Here is the question
Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
7
votes
2answers
120 views
Integrate $2\int x^2\, \sec^2x \,\tan x\, dx$
$$
2\int x^2\, \sec^2x \,\tan x\, \mathrm{d}x
$$
How to solve this using integration by parts? WolframAlpha can solve it, but is unable to give a step-by-step solution, and has a different answer to ...
1
vote
1answer
58 views
Need help integrating $\tan x$ and $\tan^n x$ using reduction
I have tried to use integration by parts taking $u$ as $\tan x$ and $v$ as $1$:
$$\int \tan x \,dx = \int \tan x \cdot 1\; dx = \tan x \cdot x - \int \sec^2 x \cdot x\; dx$$
then by taking $u$ as ...
2
votes
4answers
176 views
How do I solve $\int\frac{\cos^2(x)}{\sin(x)}\ dx$ without using Weierstass Substitution?
Every problem that I've put into wolfram alpha lately gives me instructions to substitute $\tan(\frac{x}{2})$, but I haven't been taught how to do that, nor can I understand how it works anyhow...
2
votes
1answer
94 views
How to solve $\int\cot^5x\sin^2x\ dx$?
I'm not quite sure how to approach this without it getting extremely messy... and even then, I don't know if it will come out right.
The best I can think of is to use IBP, but neither of those ...
3
votes
1answer
90 views
Integrating $\sin^2(x)$ using imaginary numbers.
I know I can change "$\sin^2\theta$" to "$\frac{1}{2}(1-\cos(2\theta))$" or use integration by parts, but I was curious about doing it using imaginary numbers. I tried this but it didn't work.
$$\int ...
2
votes
1answer
52 views
What is the value of $\int_{-1}^{1} P(\sin x)P'(\sin x)dx$?
I came across the following problem that says:
Let $P(x)=x^4+x^2+1.$ Then $\int_{-1}^{1} P(\sin x)P'(\sin x)dx=?$
My Attempt: $\int_{-1}^{1} P(\sin x)P'(\sin x)dx= \int_{-\pi}^{\pi}[\sin ...
0
votes
0answers
68 views
Finding anti derivative
It is mentioned in a different thread that $U(x)=\sin\left(\dfrac1{\ln(1+x^2)}\right)$ is an elementary function. My question is, how do you integrate it then?
2
votes
2answers
104 views
Integrate $\frac{x^2+1}{(x^2-2x+2)^2}$ using trigonometric substitutions
I've attempted to integrate the function $\frac{x^2+1}{(x^2-2x+2)^2}$ using several techniques, but none of them are solving it nicely.
I know there must be a way to do this using trig substitution ...
3
votes
2answers
72 views
Need help with this integral using trig identities
I am trying to integrate the function $$\int_{-\pi/2}^0 \sin(2x)\cos(nx) \, \mathrm{d} x.$$
My professor has an answer of $$\frac{-2\cos(\frac{n \pi}{2})+1}{n^{2}-4}.$$
When I do this problem, I ...
1
vote
0answers
46 views
Help with manipulating a change of variable in Integration
Knowing:
$$\phi (x)=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dt }{ \sqrt { 1-{ x }^{ 2 } \sin ^{ 2 }(t) } } } $$
I am trying to demonstrate that: $\phi (x)=\frac { 1 }{ 1+x } \phi \left( ...
4
votes
3answers
115 views
Help in manipulating Integrals
I try to express : $\displaystyle 1+2\sum _{ k=1 }^n \cos(2k\theta ) $
as : $\dfrac { \sin\left( \theta +2\theta n \right) }{ \sin\left( \theta \right) } $
I tried to use the exponential function ...
3
votes
2answers
112 views
The integral $\int\frac{1+\sin x}{\cos x}dx$
Is $$\int\frac{1+\sin x}{\cos x}dx$$ the same as the integral of $$\sec x+\tan x$$ (since $1/\cos x = \sec x$ and $\sin x/\cos x = \tan x$)?
2
votes
4answers
429 views
Integral of $\sin(\sqrt{x})$
I need help finding the integral of $\sin(\sqrt{x})dx$. I have the answer here but would like to know how to get there.
1
vote
2answers
64 views
Integration by substitution trig
I need to integrate $\frac {1}{2-\cos x}$ and I am given $t=\tan(x/2)$. What should I do with it?
2
votes
4answers
625 views
Find integral from 1 to infinity of $1/(1+x^2)$
I am practicing for an exam and am having trouble with this problem. Find the integral from 1 to infinity of $\frac{1}{1+x^2}$. I believe the integral's anti-derivative is $\arctan(x)$ which would ...
8
votes
0answers
179 views
Evaluating $\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$
How to integrate?
$$\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$$
I have no idea how to do it.
Tried to get some information from wiki, but its too hard :|
3
votes
0answers
158 views
Tricky integration by substitution.
I have to get this integral (EDIT: it should definitely be 1-x^2 in numerator)
$$\int_{-1}^{1} \frac{ \sqrt{1-x^2}}{1+x^{2}} dx$$
into
$$\int_{-\pi }^{\pi } \frac{1}{1+\cos^2\theta } \,d\theta - \pi$$ ...
1
vote
2answers
212 views
Integral of cosec squared ($\operatorname{cosec}^2x$, $\csc^2x$)
According to my sheet of standard integrals,
$\int \csc^2x \, dx = -\cot x + C$.
I am interested in a proof for the integral of $\operatorname{cosec}^2x$ that does not require differentiating $\cot ...
1
vote
1answer
106 views
Expressing $\int \tan^n x\,dx$ with a sum
I was playing around with integrals of $\tan x$, because I knew that both $\int\tan x\,dx$ and $\int\tan^2x\,dx$ were solvable. I then came across the fact that
$$\begin{align}
\int \tan^n x\,dx ...
5
votes
1answer
120 views
How to evaluate $\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$?
How can I integrate the following:
$$\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$$
for $a,b>0$? Maple gives a compact result:
$$\frac{1}{b} \tan^{-1}(c) - \frac{1}{ac^2} ...
4
votes
1answer
215 views
How can I compute $\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$
If $f(x)=\text{arccot}(x)$ for non-negative $x$ and $0$ otherwise, how can I calculate
$$\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$$
for $y\in\mathbb{R}$?
6
votes
1answer
175 views
Verifying $\int_0^\pi \sin(x) /2(\sin(x/(2n+1)) \,dx \leq \pi$
I'm having trouble verifying this inequality. It goes like this (appears in Giaquinta, Mathematical analysis, linear and metric structures, page 445):
$$
\int_{0}^{\pi} ...
11
votes
2answers
353 views
$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \mathrm {d}x$ Evaluate Integral
Here is a fun integral I am trying to evaluate:
$$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$
I thought about integrating by parts $2n$ times and then using ...
1
vote
3answers
91 views
Find the indefinite integral of $1/(16x^2+20x+35)$
Here is my steps of finding the integral, the result is wrong but I don't know where I made a mistake or I may used wrong method.
$$
\begin{align*}
\int \frac{dx}{16x^2+20x+35}
&=\frac{1}{16}\int ...
2
votes
2answers
64 views
Evaluating integral similar to $\sin^{-1}(3x)$
I was just doing revision for an upcoming exam and I came across a question I do not know how to answer. It seems pretty simple, however I am having a blank moment. If anyone knows how to solve that I ...
3
votes
2answers
83 views
Show $\int_\frac{1}{3}^\frac{1}{2}\frac{\operatorname{artanh}(t)}{t}dt=\int_{\ln 2}^{\ln 3}\frac{u}{2\sinh u}du$
How would I show (or explain) that
$$\int_\frac{1}{3}^\frac{1}{2}\frac{\operatorname{artanh} t}{t}dt,$$
$$\int_{\ln 2}^{\ln 3}\frac{u}{2\sinh u}du,$$
and
$$-\int_\frac{1}{3}^\frac{1}{2}\frac{\ln ...
12
votes
0answers
651 views
Computing ${\int\limits_{0}^{\chi}\int\limits_{0}^{\chi}\int\limits_{0}^{2\pi}\sqrt{1- \cdots} \, d\phi \, d\theta_1 \, d\theta_2}$?
This question led me to this integral I can not solve:
$$
...
1
vote
1answer
196 views
Integrate $(\cos x) ^ 4$
Integrate $(\cos x)^4$. I see solutions using power reduction everywhere. I vaguely remember doing it based on some manipulation of trig identities $(\cos x)^2 = 1 - (\sin x)^2$ and $u$-substitution ...
3
votes
1answer
258 views
Integration Problem Proof ($\sin x$)
Problem: Integration of $\displaystyle\int_{-1}^1 {\sin x\over 1+x^2} \; dx = 0 $
(according to WolframAlpha Definite Integral Calculator)
But I don't understand how. I tried to prove using ...
4
votes
1answer
268 views
Evaluate $\int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx$
How to evaluate
$$ \int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx $$
I don't know how to deal with it.
0
votes
3answers
133 views
Integral of a trigonometric function [duplicate]
Possible Duplicate:
Evaluating $\int P(\sin x, \cos x) \text{d}x$
How do I integrate the following function?
$$\frac{\sin 2x}{(1 + \cos^2x)^2}?$$
Thanks.
1
vote
3answers
142 views
Integration: How to Begin? [duplicate]
Possible Duplicate:
Help evaluating $\int \frac{dx}{(x^2 + a^2)^2}$
How to I begin this integration problem?
$\begin{align}\int_{0}^{1} \frac{dx}{{\left(x^2 + 1\right)}^{2}}\end{align}$
...
3
votes
2answers
239 views
Find $\int e^{2\theta} \cdot \sin{3\theta} \ d\theta$
I am working on an integration by parts problem that, compared to the student solutions manual, my answer is pretty close. Could someone please point out where I went wrong?
Find $\int e^{2\theta} ...
2
votes
2answers
128 views
Methodology for Integration by Parts
I am looking at an example of integration-by-parts in my Calculus book, and there is one thing that I do not understand:
Prove the reduction formula: $$\int \sin^n x \ dx = -\frac{1}{n} \cdot \cos x ...
2
votes
3answers
114 views
Evaluating an Trigonometric Integral
I'm completely at a loss on how to solve this integral. I've tried to put it in programs like Wolfram Alpha and Microsoft Maths to get some semblance of an answer to aim for but all I get is a garbled ...
2
votes
2answers
690 views
Integrate $\csc^3{x} \ dx$
I found these step which explain how to integrate $\csc^3{x} \ dx$. I understand everything, except the step I highlighted below.
How did we go from:
$$\int\frac{\csc^2 x - \csc x \cot x}{\csc x - ...
0
votes
2answers
1k views
Using double angle formulas in integration, trouble following an example.
I have just started looking at integration and I am having trouble understanding what has been done in one of the examples in the book I am working through.
It involves using the double angle ...
3
votes
1answer
440 views
Integrating $\int \sin^n{x} \ dx$
I am working on trying to solve this problem:
Prove: $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot x \ \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$
Here are the steps that I ...
3
votes
1answer
283 views
Integration by Parts with Trigonometric Functions
Trying to evaluate this indefinite integral:
$$ \int (x^2 + 1)\cos2xdx$$
So far I have the following: $u=x^2 + 1 \Rightarrow du = 2xdx$ and $dv=\cos2x \Rightarrow v = \frac {\sin2x}{2}$. So the ...
