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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 } ...