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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
1answer
53 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{\...
4
votes
2answers
83 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{...
0
votes
2answers
84 views

How can I prove $\int_0^{\pi/2} \tan ^nx \,{\rm d}x=\frac{\pi}{2}\sec{\frac{n\pi}{2}}$?

How can I prove $\displaystyle\int_0^{\pi/2} \tan ^nx \,{\rm d}x=\frac{\pi}{2}\sec{\frac{n\pi}{2}}$? Do I need to use any special function to solve this integration?
2
votes
1answer
46 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: $\...
4
votes
2answers
319 views

find $\int^{\pi/2}_0 \frac{\sin^{n-2}(x)}{(1+\cos x)^n}dx$

Finding $$\int^{\pi/2}_0 \frac{\sin^{n-2}(x)}{(1+\cos x)^n}dx$$ what i try $$ \begin{split} I &= \int^{\pi/2}_0 \left(\frac{\sin x}{1+\cos x}\right)^n\csc^2(x)dx \\ &= \int^{\pi/2}_0 \...
2
votes
1answer
67 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{\...
-3
votes
0answers
33 views

Evaluating $\int_0^\infty \frac{\cos(ax)}{\cosh(b)+\cosh(x)} dx$ [closed]

How to integrate this function? $$\int_0^\infty \frac{\cos(ax)}{\cosh(b)+\cosh(x)} dx$$ where $a$ is real and $b>0$ I put this problem in my integral calculator(online) and in WolframAlpha ...
0
votes
1answer
31 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
22 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
68 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
36 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
231 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
37 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
31 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
57 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 $$\int \cosh^5(x)-\cosh^3(x) dx$$ ...
-1
votes
0answers
17 views

Volume of a body with rotation between y=cos(x) and x axis

I have a math problem, can you help me? I need to find volume of a body with rotation between $y=\cos(x)$ and $x$-axis on interval $[0,3\pi /4]$ about $y=0$. I need to draw graph, find the bounds and ...
3
votes
3answers
81 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
17 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
87 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
119 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
135 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
65 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
95 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
80 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
97 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
64 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
238 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
70 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 } ...
1
vote
1answer
84 views

$ \int \bigl| a\sin(x)+b\cos(x) \bigr| {\rm d}x = ? $

My friend evaluated this to be $$ \int \bigl| a\sin(x)+b\cos(x) \bigr| {\rm d}x \\ = \sqrt{a^2+b^2} \left( \sin(x-\phi)\text{sign}(\cos(x-\phi)) +\frac{2}{\pi} \bigl(x-\arctan(\tan(x-\phi)) \bigr) \...
0
votes
2answers
72 views

Prove $\int_0^\infty\operatorname{sech} x\,dx=\pi/2$, and deduce $\int_0^1\operatorname{sech}^{-1}x\,dx$

Prove $$\int_0^\infty\operatorname{sech} x\,dx=\pi/2$$ and deduce $$\int_0^1\operatorname{sech}^{-1}x\,dx$$ I can prove the first statement (see below), but I was unable to deduce the value of the ...
1
vote
1answer
77 views

Is there any solution (numeric or closed) to integration of ${\sin x}^{\cos x}$?

I've tried so many ways to evaluate $\int{\sin x}^{\cos x}dx$ and even searched and used programs like matlab, maple and scipy library and got no answer! my question is clear, is there any numerical ...
4
votes
4answers
147 views

A Difficult Definite Integral $\int_0^{2\pi}(t-\sin t)(1-\cos t)^2{\rm d}t.$

Problem Evaluate $$\int_0^{2\pi}(t-\sin t)(1-\cos t)^2{\rm d}t.$$ Comment It's very complicated to compute the integral applying normal method. I obtain the result resorting to the skillful formula ...
1
vote
0answers
51 views

Proving an trigonometric-integral inequality [closed]

I'm interested in any methods that can be used to solve: $$\int_{0}^1 \left[1+\sin \left(\frac{\pi}{2}x\right)\right]^n dx \gt \frac{2^{n+1}-1}{n+1} \quad (n=1,2,...)$$ At this point, I'm unsure how ...
2
votes
1answer
74 views

$\int\limits_{0}^{2\pi}[\int\limits_{0}^{\pi}\sin^3\theta \cos^2\{r(\sin \theta \sin \alpha\cos \phi+\cos \theta\cos \alpha)+c\}d\theta] d \phi$

Could you tell me how to compute the following definite integration? $$\int\limits_{\phi=0}^{\phi=2\pi}\int\limits_{\theta=0}^{\theta=\pi}\sin^3\theta \cos^2\{r(\sin \theta \sin \alpha\cos \phi+\cos \...
5
votes
1answer
97 views

Evaluating $ \int \frac1{5S_{8} - 9S_{10} + 7S_{12} - 2S_{14}} {\rm d}x$, where $S_n = \sin^n(x) + \cos^n(x)$

Prove that $$ \int \frac1{5S_{8} - 9S_{10} + 7S_{12} - 2S_{14}} {\rm d}x = 2x - \arctan \left( \frac{\tan2x}{2 + \tan^22x} \right) + C$$ where $S_n = \sin^n(x) + \cos^n(x)$. Even ...
1
vote
2answers
57 views

Why $\int_0^{2\pi} \sin^2 x \mathrm{d}x \neq \int_0^{2\pi}\sin x \sin nx \mathrm{d}x \to n=1$

By orthogonality, we know that $$\int_0^{2\pi}\sin mx \sin nx \mathrm{d}x = \pi$$ iff $m=n$ and $0$ otherwise. Nevertheless, when I am calculating it as follows, I get $0$. $$\int_0^{2\pi} \sin x \...
5
votes
2answers
92 views

Integral of $\int\frac{\sin x}{\sin x+\cos x}dx$

The questions defines $$I=\int\frac{\sin x}{\sin x +\cos x}dx\;\;J=\int\frac{\cos x}{\sin x +\cos x}dx$$ It asked me to find $I+J$ and $J-I$ which I have done and I will show below but now I need to ...
2
votes
2answers
69 views

Integrating $\int \sec xdx$: Why is $\ln|\text{sec}x + \text{tan}x| + C$ preferred over $\tanh^{-1}(\sin x) + C$?

I was trying to integrate $\sec^3x$ and discovered that I would have to integrate $\sec x$ in the process. I had not seen the "standard" approach and came up with my own solution, which is apparently ...
0
votes
0answers
41 views

Integral $\cos\left(\ln\left(e^t+1\right)+at+b\right)dt$

I am trying to solve this integral: $$\int \cos\left(\ln\left(e^t+1\right)+at+b\right)dt $$ with $$a, b \in \Re$$ I can solve this: $$\int \cos\left(\ln\left(x\right)\right)dt $$ Defining: $u=\cos(...
0
votes
3answers
60 views

How do I integrate $\sqrt{1+\sec2x}$? [closed]

I tried converting $\sec 2x$ into $1/\cos 2x$, and then performing further simplification which led me to $$\int \sqrt{\frac2{1-\tan^2 x}} dx$$ I'm unable to take it from here.
4
votes
4answers
104 views

Prove that $\int_0^{\frac{n\pi}{4}} \frac{1}{3\sin^4(x) + 3\cos^4(x) -1} dx = \frac{n\pi}{4} $

Prove that $$\int_0^\frac{n\pi}{4} \left(\frac{1}{3\sin^4(x) + 3\cos^4(x) -1}\right) dx = \frac{n \pi}{4} $$ is true for all integers $n$. Here I've looked at the case where $n=1$ as $t=\tan(x)$ is ...
0
votes
1answer
37 views

A trigonometric function question for calculus [closed]

Compute the definite integral $$\int_0^1 \left(2-x^2\right)^{3/2}dx$$ Please help me, i suppose $x=\sqrt{2} \sin \theta$ but I just couldn't get the answer. Thank you
0
votes
1answer
31 views

Calculation of $\int_{-R}^R(R-x^2)^\frac{n-1}{2}dx$

Let $R>0$ be fixed, $n\ge 2$. What is $$\int_{-R}^R(R-x^2)^\frac{n-1}{2}dx?$$ I tried substitution $x=\sqrt{R}\sin(t)$. Then $dx=\sqrt{R}\cos(t)dt$ and $$\int_{-R}^R(R-x^2)^\frac{n-1}{2}dx=\int_{\...
2
votes
1answer
59 views

Calculate the integral of exp(z)/sin(z) over |z|=4 using the residue theorem.

Click to view the integral in correct format. Calculate the integral of exp(z)/sin(z) (as in the image above) over the positively oriented circle defined by |z|=4 using the residue theorem. This is ...
0
votes
0answers
43 views

About $ f(w,L) = \int_1^w \int_0^{2 L \pi} \frac{ \ln(\frac{\sin(x) +\ sin(vx)}{2} + \frac{5}{4})}{L(w - 1)} dx dv $

Consider $$ f(w,L) = \int_1^w \int_0^{2 L \pi} \frac{ \ln(\frac{sin(x) + sin(vx)}{2} + \frac{5}{4})}{L(w - 1)} dx dv $$ For real $w > 1 $ and integer $ L > 1$ Conjecture : $$ \lim_{L \to \...
0
votes
3answers
56 views

Can these integrals be solved by u-substitution using trig identities?

$\int \frac{\operatorname{sin} x}{2\sin x\cos x+5}dx$ $\int \frac{\operatorname{sin}x\cos x}{\operatorname{sin}x+\cos x}dx$ $\int \frac{dx}{\operatorname{sin} x+\cos x}$ I'd like to know if there's ...
2
votes
2answers
44 views

Contour intergal of a rational trigonometric function, can't find my mistake.

This is the integral : $$I = \int_{0}^{2\pi} \frac{dx}{(5+4\cos x)^2}\ $$ Which, according to wolfram alpha, should evaluate to $\frac{10\pi}{27} $, but the value i find is $\frac{20\pi}{27} $. These ...
0
votes
0answers
57 views

Trigonometric Integral Involving the fractional part

Let $\{\}$ denote the fractional part function, does the following Integral admit a closed-form ? $$\int_{0}^{\pi/2}\bigg\{\frac{1}{\cos(x)}\bigg\}dx$$