Questions tagged [trigonometric-integrals]
Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.
-2
votes
2answers
56 views
The Integral of $\int \sin(ax) \cos(ax) dx$
What is the integral of:
$$I=\int \sin(ax) \cos(ax) dx$$
My approach is down below. I have attempted the problem and posted it as an answer. I did the problem using trigonometric substitution.
$$u=...
2
votes
3answers
92 views
Evaluate the trigonometric integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin^2\theta \cos^2\theta}{(\cos^3 \theta+\sin^3\theta)^2} \, \mathrm{d}\theta$ [closed]
$$\int_{0}^{\frac{\pi}{2}} \frac{\sin^2\theta \cos^2\theta}{(\cos^3 \theta+\sin^3\theta)^2} \, \mathrm{d}\theta$$
Please someone help me with this integral, the hint was to use $t=\tan\left(\frac{\...
2
votes
1answer
77 views
Series of Beta Function
Solve:
$$\int_0^\frac{\pi}{2}\frac{\sqrt{\sin{x}\cos{x}}}{\cos{x}+1}dx$$
I tried
$$\int_0^\frac{\pi}{2}\frac{\sqrt{\sin{x}\cos{x}}}{\cos{x}+1}dx=\sum_{k=0}^\infty(-1)^k\int_0^\frac{\pi}{2}(\sin{x})^...
1
vote
2answers
30 views
show $\int \frac{1}{\cosh(x)}dx = \arctan(\sinh(x))$ using the substitution $u=\sinh(x)$
I am trying to show that $$\int \frac{1}{\cosh(x)}dx = \arctan(\sinh(x))$$ Using the substitution $u=\sinh(x)$
So if $u=\sinh(x)$, then $$\frac{du}{dx}=\cosh(x)$$ thus
$$\int \frac{1}{\cosh(x)} \...
1
vote
1answer
82 views
Evaluation of $\int_{1}^{\infty} \frac{\arctan (x+1)}{x^2+4} dx$ in closed form? [duplicate]
I have used the substitution $u= x+1 $ then $du =dx$ to evaluate the following integral in closed form and since $\arctan$ is connected to $x^2+1$ as it is a derivative of it.
$$\int_{1}^{\infty} \...
3
votes
0answers
60 views
Definite integral of the sum of quadratics in sine and cosine raised to a negative power
I have a definite integral that arises when looking at Lennard-Jones potentials involving ellipses. It is of the form $$\int_{0}^{2\pi}\frac{f\left(\sin\left(t\right),\cos\left(t\right)\right)}{\left(...
0
votes
1answer
28 views
Trigonometric integrals involving tangent
So I came across a problem after answering the integral. The problem was:
$\int\tan^3(3x)dx$. This is to be integrated. This is how I did it:
$$\begin{align}\int\tan^2(3x)\tan(3x)dx&=\int(\sec^2-...
0
votes
2answers
68 views
Prove $\int^{\infty}_{0} \frac{\cos(ax)}{\cosh(\beta x)}dx = \frac{\pi}{2\beta}\operatorname{sech}(\frac{a\pi}{2\beta})$
Proof of 3.981.3 Gradshteyn ed.8.
$$\int^{\infty}_{0} \frac{\cos(ax)}{\cosh(\beta x)}dx = \frac{\pi}{2\beta}\operatorname{sech}(\frac{a\pi}{2\beta})$$
I was interested in the derivation (not ...
0
votes
1answer
46 views
How do I justify the convergence of :$\int_{-\infty}^{+\infty}\dfrac{\exp(-x^{2n+1}\arctan (x))}{1+x^{2n}}dx$ with $n$ a positive integer?
This integral : $$\int_{-\infty}^{+\infty}\dfrac{\exp(-x^{2n+1}\arctan (x))}{1+x^{2n}}dx$$ seems converge for $n$ is a natural number according to some values of $n$ which I run in wolfram alpha , My ...
0
votes
2answers
56 views
Integrate $\int \frac{\cos^2 x}{(1 - \cos x)\sin x} dx$
Integrate $\int \frac{\cos^2{x}}{(1 - \cos{x})\sin x} dx$
So far I have gotten here: $-\int -\frac{\cos^2 (x) \sin x}{(\cos (x) - 1)(\cos^2 (x)-1)} dx$
here I can substitute $u = \cos x$ , $ -\...
1
vote
0answers
91 views
Closed form solution to $\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$
Is there closed form solution to this integral
$$\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$$
$r(\theta)=\frac{a}{\cos(\theta)}$ is radius vector from the ...
0
votes
1answer
54 views
Does $-\frac12\cos(2x)=\sin^2(x)$? [duplicate]
$$\int2\sin x\cos x\mathrm dx$$
$1$:Consider $2\sin x\cos x=\sin(2x)$ Then the integration is $-\frac12\cos(2x)$
$2$: ...
0
votes
0answers
36 views
Solution to an equation involving integrals
I basically have a integral equation which I then want to put on a lattice. I want to find a continuous and periodic function $F(q^1,q^2):\mathbb{R}^2\rightarrow\mathbb{R}$ which solves
$$\int_0^1F(q^...
1
vote
0answers
62 views
Proof that $\frac{2\times2\times2\times4\times4\cdots(2n)\times(2n)}{3\times3\times5\times5\cdots(2n-1)\times(2n+1)}$ converges to $\frac{\pi}{2}$
I defined
$$S_{2n}=\frac{1\cdot 3\cdot 5\cdots (2n-3)\cdots (2n-1) }{2\cdot 4\cdot 6\cdots (2n-2)\cdot (2n)}\cdot \frac{\pi}{2}$$
and
$$S_{2n+1}=\frac{2\cdot 4\cdot 6\cdots (2n-2)\cdots (2n) }{1\cdot ...
7
votes
2answers
439 views
Trig Subsitution When There's No Square Root
I would say I'm rather good at doing trig substitution when there is a square root, but when there isn't one, I'm lost.
I'm currently trying to solve the following question:
$$Ar \int_a^\infty \frac{...
3
votes
2answers
48 views
solve $\int_{-\sqrt{3}}^{\sqrt{3}} 4 \sqrt{3-y^2}dy$
$\int_{-\sqrt{3}}^{\sqrt{3}} 4 \sqrt{3-y^2}dy$
trig sub
$y = \sqrt{3}\sin(u)$
$dy = \sqrt{3}\cos(u)du$
\begin{align}\int_{\sqrt{3}\sin(-\sqrt{3})}^{\sqrt{3}\sin(\sqrt{3})}& 4 \sqrt{3-3\sin^2(u)...
0
votes
1answer
69 views
Prove $\int_0^\pi\frac{\cos(n\theta)}{\cos\theta - \cos\theta_0}d\theta = \frac{\pi\sin(n\theta_0)}{\sin\theta_0}$
I ran across the following formula in a textbook but can't figure out how to prove it. How would I go about solving this?
$\int_0^\pi\frac{\cos(n\theta)}{\cos\theta - \cos\theta_0}d\theta = \frac{\pi\...
0
votes
0answers
14 views
Trig Vector Problem- Finding How fast the wind blows
A planes actual resulting ground speed and direction are 550 mph and N26E. A wind is blowing from the northeast. If the pilot is holding a course of N27E how fast is the wind blowing?
This question ...
0
votes
0answers
65 views
Expressing $\int_{0}^{2\pi} e^{ikr(\sin\alpha\cos\beta-\sin\theta\cos(\phi-\beta))}\; d\beta$ as a Bessel function
How to express the trigonometric function
$$\sin\alpha\cos\beta-\sin\theta\cos(\phi-\beta)$$
inside the integral to become Bessel function?
$$I=\int_{0}^{2\pi} e^{ikr(\sin\alpha\cos\beta-\sin\theta\...
0
votes
3answers
79 views
Prove $\int_{-\pi}^{\pi} \frac{dk}{2 \pi} \frac{\cos(nk)-1}{\cos(k)-1} = n$
The following integral comes up in the theory of Brownian motion in 1D:
$$\int_{-\pi}^{\pi} \frac{dk}{2 \pi} \frac{\cos(nk)-1}{\cos(k)-1} = n$$
where $n \in \mathbf{N}$. Does anybody know a short ...
4
votes
3answers
292 views
Find the smallest value of $f(x) := \left({1\over9}+{32\over \sin(x)}\right)\left({1\over32}+{9\over \cos(x)}\right)$ on the interval $(0,\pi/2)$
There's a function defined as:
$$f(x) := \left({1\over9}+{32\over \sin(x)}\right)\left({1\over32}+{9\over \cos(x)}\right)$$
In interval $$\left(0,\frac{\pi}{2}\right)$$
Find the smallest value (...
3
votes
1answer
68 views
u-sub vs trig-sub are giving different answers for $\int\frac{x+4}{x^2+2x+5}dx$
When I complete the square in the denominator and solve using u-sub, I can get the right answer:
$$\int\frac{x+4}{x^2+2x+5}dx$$
$$\int\frac{x+4}{x^2+2x+1+4}dx$$
$$\int\frac{(x+1)+4}{(x+1)^2+4}dx$$
$$...
2
votes
0answers
96 views
Integral of $\int_0^b \cos(x)\cos(\frac a x)dx$
I've tried integration by parts. I can integrate both factors:
$\int\cos(x)dx = \sin(x) + C_1$
$\int\cos(\frac a x) = a Si(\frac a x) + x \cos(\frac a x) + C_2$
However I'm stuck at this point. I ...
2
votes
1answer
58 views
Confusion regarding a definite integral in a solution
While following this solution for warping functions, there is a part that I would need some clarification on. The author has stated that using orthogonality of terms in the sine series $\int_{-a}^a{\...
5
votes
3answers
106 views
Prove that, $\int_{0}^{2\pi}\frac{\cos x+2}{5+4\cos x} dx=\pi$
I have tried to solve this which goes as follows-
$$\begin{align*}
\int\frac{\cos x+2}{5+4\cos x} dx
&=\int\frac{(1/4)(5+4\cos x)+(3/4)}{5+4\cos x} dx\\
&={1\over4}\int dx+{3\over4}\int\frac{...
1
vote
1answer
60 views
The limit of the maximum of a sum of sines
I've recently stumbled upon the following problem from Brilliant:
Compute the following:
$$\lim_{n\to\infty}\max_{x\in[0,\pi]}\sum_{k=1}^n\frac{\sin(kx)}k$$
Options:
$\...
7
votes
3answers
366 views
Find $\int^{\pi/2}_0 \frac{\sin^{n-2}(x)}{(1+\cos x)^n}\mathrm dx$
Finding
$$I=\int_0^{\pi/2}\frac{\sin^{n-2}(x)}{(1+\cos x)^n}\mathrm dx$$
What I tried:
\begin{align*}
I &= \int^{\pi/2}_0 \left(\frac{\sin x}{1+\cos x}\right)^n\csc^2(x)\mathrm dx \\
&=...
2
votes
1answer
72 views
Integration of $\sqrt{\sec x}$.
This problem is a challenge question given to me by my friend. He asked me to find
$$\displaystyle{I=\int\sqrt{\sec x}}\ dx$$
What I tried:
Let $\sec x=u$. So $$du = \sec x\tan x\ dx=\sec x\sqrt{\...
0
votes
1answer
37 views
Explicit formula for $\int \text{sinc}(2 \pi x)^{2 k-1} \, \mathrm dx$?
This is very similar to other questions people have asked (I've Googled extensively), but not identical. I have been trying to use integration by parts to generate an explicit formula in terms of $x$ ...
0
votes
1answer
24 views
How to understand the definition of an inverse substitution when finding the primitive function?
From textbook:
Inverse substitutions: Let $f$ be a function defined on an interval $I$. Let $g$ be a function from an interval $J$ into an interval $I$ which is differentiable on $J$, and let $h$ ...
2
votes
3answers
79 views
How to calculate $\int_{0}^{\pi/3} \sqrt{\sec ^2 (x)} dx$?
I'm trying to calculate the following integral:
$\int_{0}^{\pi/3} \sqrt{\sec^2 (x)} dx$
But I have no idea where to start. Can you give me some advice?
1
vote
1answer
45 views
$\int_{0}^{1} t^2 \sqrt{1+4t^2} dt$ plugging in limits assist
Solve $\int_{0}^{1} t^2 \sqrt{1+4t^2} dt$
Making the substitution $t=\frac{\sinh(x)}{2},$ then $4t^2=\sinh^2(x),$ and we get $dt=\frac{\cosh(x)}{2}dx$
$$\begin{align}
\int t^2 \sqrt{(1+4t^2)}dt&=...
11
votes
2answers
264 views
Solving $\int_0^{\infty} \ln^m(x)\sin\left(x^n\right)\:dx$
Spurred on by this question, I decided to investigate a more generalised form:
\begin{equation}
I_{m,n} = \int_0^{\infty} \ln^m(x)\sin\left(x^n\right)\:dx
\end{equation}
Where $n,m \in \mathbb{N}$
...
0
votes
1answer
40 views
Indefinite Integration of a trig function
How do you integrate
$$
1/(\sin2x(\tan^5x+\cot^5x))
$$
with respect to $x$?
I tried writing tan and cot in terms of sin and cos but when I take $\mathrm{LCM}$ I get powers of $10$ for $\sin$ and $\...
1
vote
3answers
33 views
$\int\frac{\sin(x)}{\sqrt{1-\cos^2(x)}}$ Where did I mess up the domain?
So we have $$\int\frac{\sin(x)}{\sqrt{1-\cos^2(x)}}dx$$
My book uses a $u$-substitution with $u=\cos(x)$, $du=-\sin(x)$,and they get $$\int\frac{-du}{\sqrt{1-u^2}}$$ which gives them $\arccos(u)+C=\...
1
vote
3answers
67 views
Evaluate $\int \cosh^3 (x) \sinh^2 (x )dx $
Evaluate $$\int \cosh^3 (x) \sinh^2 (x )dx $$
So my original thought was to apply the identity that $\sinh^2(x)=\cosh^2(x)-1$. This means that my integral becomes
$$\int \cosh^5(x)-\cosh^3(x) dx$$
...
3
votes
2answers
100 views
I'm stuck integrating $\int \sqrt{x^2-a^2} dx$ using trigonometric substitution
When I'm trying to integrate $\int \sqrt{x^2-a^2} dx$ using trigonometric substitution, I get stuck. Here's the complete solution so far:
$$
x(\theta)=a\sec{\theta}\\
x'(\theta)=a\tan{\theta}\sec{\...
0
votes
0answers
23 views
Using Area of Segment - Derive General Formula for the Volume of a Tilted Cylinder Partially Filled with Water
I would be very grateful if I could get some advice as to where I am going wrong! I am trying to derive the formula to calculate the volume of a partially filled cylinder and it just doesn't seem to ...
4
votes
6answers
94 views
What's the answer to $\int \frac{\cos^2x \sin x}{\sin x - \cos x} dx$?
I tried solving the integral $$\int \frac{\cos^2x \sin x}{\sin x - \cos x}\, dx$$ the following ways:
Expressing each function in the form of $\tan \left(\frac{x}{2}\right)$, $\cos \left(\frac{x}{2}\...
4
votes
3answers
123 views
How to integrate $\int_{0}^{2\pi}\frac{\mathrm dx}{(1-a\cos x)^2}$
I'm having a hard time trying to resolve this integral :
$$\int_{0}^{2\pi}\frac{\mathrm dx}{(1-a\cos x)^2}$$
where $a$ is a positive real constant.
I tried using substitution, but I'm stuck by the ...
4
votes
2answers
136 views
Is there a closed form for the trigonometric integral $\int\limits_0^{\pi/4}\frac{\cos(2k+1)x}{\cos x} dx$?
One can easily show that $\int\limits_0^{\pi}\frac{\cos(2k+1)x}{\cos x} dx = 2 \int\limits_0^{\frac{\pi}{2}}\frac{\cos(2k+1)x}{\cos x} dx = (-1)^k \pi$.
But is there a closed form for $\int\limits_0^...
0
votes
2answers
60 views
Trig substitutions. What is going on?
I am a bit confused about my book's explanation regarding trig substitutions. I understand u-subs... but I don't get this:
first where did the equation $x = a \sin\theta$ come from? What is x here? ...
0
votes
1answer
40 views
The integral of
I have to take the indefinite integral of the following function:
$$\int_\limits{0}^{\frac{\pi}{2}}\sin^2(\frac{1}{3}\theta)d\theta$$
I did a double substitution, my first was:
$$3\int_\limits{0}^\...
1
vote
2answers
67 views
Not getting the right answer with alternate completing the square method on $\int\frac{x^2}{\sqrt{3+4x-4x^2}^3}dx$
So I've looked up how to do this problem and when they complete the square it's using the $(\frac{-b}{a})^2$ method where, in the denominator, they factor out the $4$, and then make:
$$x^2-x=x^2-x+{1\...
0
votes
1answer
106 views
Integration by Parts only of $\sqrt{1-u^2}$
I am trying to integrate the function:
$$f(x)=\sqrt{1-u^2}$$
I was using integration by parts to attack the problem, and it was:
$$\int\sqrt{1-u^2}du$$
I set $g=\sqrt{1-u^2}$ and $dv=du$
Thus leading ...
2
votes
3answers
82 views
Don't understand how to use trig sub on $\int\frac{x^3}{(4x^2+9)^\frac{3}{2}}dx$
In my textbook it says
First we note that $(4x^2+9)^\frac{3}{2} = (\sqrt{4x^2+9})^3$ so trigonometric substitution is appropriate. Although $\sqrt{4x^2+9}$ is not quite one of the expressions in ...
3
votes
1answer
102 views
What is the solution for $\int_{0}^{\pi/2}\frac{\cos^2x}{\cos^2x+4\sin^2x}\,dx$
I used the rule :$$\int_{0}^{a}f(x) = \int_{0}^{a}f(a-x)\,dx$$
And got :$$\int_{0}^{\pi/2}\frac{\sin^2x}{\sin^2x+4\cos^2x}\,dx$$
Then, I added both the question and the above integrand, and I got :$$\...
2
votes
2answers
66 views
Integral of $\sec(x)$ using $u$ sub
I've just begun learning how to integrate and I wanted to see if I could integrate $\sec(x)$ by $u$-substitution. After getting my answer, I was told it couldn't be in complex form, but why, and if so,...
11
votes
1answer
253 views
Solving $\int_0^{\frac{\pi}{2}}\frac{1}{\sin^{2n}(x) + \cos^{2n}(x)}\:dx$
Spurred on this question I decided to investigate the following integral:
\begin{equation}
I_n = \int_0^{\frac{\pi}{2}}\frac{1}{\sin^{2n}(x) + \cos^{2n}(x)}\:dx
\end{equation}
Where $n \in \mathbb{...
1
vote
3answers
71 views
Integration of powers of trigonometric function with linear term
I got stuck trying to find a general formula for the following integral
$$\int_0^{\pi} t \cdot\cos^{2n}{\left(\frac{t}{2}\right)} \, dt =
4 \int_0^{\pi/2} t \cdot\cos^{2n}t \,dt \; , \; \text{ for } ...
