1
vote
1answer
29 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
165 views

Fun 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
129 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
52 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
37 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?
3
votes
2answers
32 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
41 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}} \, ...
0
votes
0answers
20 views

area of surface Function for Champagne flute

Can any body tell me how to solve number 5 on this page i have attached, finding a function for champagne flute while function for wine glass is given.
0
votes
0answers
35 views

Stuck with trig substitution

I am stuck with problem at my homework assignment. $$\int \sqrt{1+4x^2}dx$$ I try to apply trigonometric substitution $$x = \frac 1 2\tan{2u}$$ $$dx = \frac 1 {\cos^2{2u}}du$$ But after ...
1
vote
1answer
39 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
43 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 ...
1
vote
0answers
20 views

Trigonometry integration with a bound

So, I want to integrate $\int_\gamma sinz\; dz$ where $\gamma$ is any curve joining $i\to \pi$. Can I say that it is beacause $\int sinz=-cosz$, and $-cosz$ is analytic on the domain containing ...
1
vote
2answers
54 views

Find the area of intersection determined by three circles (Green's Thm)

I'm looking to find the shared area between these three circles using Green's Theorem: $$x^2+y^2=1$$ $$(x-1)^2 + y^2 = 1$$ $$\left(x-\frac{1}{2}\right)^2 + \left(y - \frac{\sqrt{3}}{2}\right)^2 = 1$$ ...
0
votes
4answers
78 views

Why is $\int_{0}^{2\pi} |\sin x| dx = 4$

I can't understand why $$\int_{0}^{2\pi} |\sin x| dx = 4$$while $$\int_{0}^{2\pi} \sin x dx = 0$$ I did the calculus for the second varian but I can't reach result $4$ for the first integral. Thank ...
14
votes
8answers
2k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
3
votes
1answer
36 views

Evaluating trigonometric integral and Cauchy's Theorem

I am trying to evaluate the following integral: $\int_0 ^\pi {d\theta\over{1+\sin^2\theta}}$ I tried using the substitution of $\sin\theta={1\over 2i}(z-1/z)$, where $z=e^{i\theta}$, and ...
1
vote
1answer
38 views

Hairy $u$-substitution problem

Evaluate using the $u$-substitution: $$\int_{-1}^{1} \frac{dx}{4 + x^2}.$$ Now, I was told to set $$\tan u = \frac{x}{2},$$ but that doesn't help me at all. Hints needed!
8
votes
2answers
76 views

Integrating $\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$

I am a little bit lost with integral: $$\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$$ I have already worked on in and done substitution $x = \sin(t)$: This brings me to: ...
1
vote
1answer
70 views

Evaluation of the integral $\int_{-6}^{-3}\frac{\sqrt{x^2-9}}{x}$

How to evaluate the following integral? I have tried the following things but I have no idea to continue after the last step. Moverover, the integral seems wrong when compared with the ans from ...
2
votes
2answers
81 views

Trigonometric Inequality $\cos 1 +\cos2+\ldots +\cos n < 0.55$ can be solved with the help of Integrals?

How can I prove for every $n \in \mathbb{N}$ $$\cos 1 +\cos2+\ldots +\cos n < 0.55$$ Any idea, any solution? Thanks! EDIT Can be solved this inequality with the help of integrals, because I met ...
0
votes
1answer
20 views

Understanding less trivial integration by trig substitution.

In less trivial integration problems, using trig substitution (1.1), I keep fumbling over when and what trig identities to use. One trig identity I need help proving/understanding is how/why ...
1
vote
0answers
45 views

Integration Question - Not sure how to approach

I have absolutely no idea how to approach this question: $$\int \frac{x^2}{(15+6x-9x^2)^{3/2}} \ \mathrm{d}x$$ I'm almost positive that it has something to do with trigonometric substitution, but ...
0
votes
2answers
36 views

how to get $2/(t^2 + 1)$ as the derivative for Sin(theta) when $\tan(\theta/2) = t$

If $\sin \theta = \frac{2t}{1 + t^2}$ How do you get $d\theta = \frac{2}{1 + t^2}$ If you differentiate by quotient rule you get $\frac{2(1 - t^2)}{(1+t^2)^2}$ It is part of the solution to ...
2
votes
1answer
93 views

Integration of $1/(1+\sin x)$

I solved it using $t=\tan(\frac{x}{2})$ substitution and got $-2/(1+\tan(x/2))+C$, but in my math book solution is $\tan(x/2-\pi/4)+C$. Are those the same expressions and if they are, how do I ...
-1
votes
0answers
25 views

Finding the area under an arcsecant curve?

this question was in my revision, and I've been having a little bit of trouble with it. "Find the volume of the solid formed if the curve $y=sec^{-1}x$ is rotated about the x-axis from x=0 to x=0.5, ...
4
votes
2answers
118 views

Find the integral $\int \frac{1}{x^2 \cdot \tan(x)} \ dx$

This problem seems pretty tricky. I need to find the integral of $$\int \dfrac{1}{x^2 \cdot \tan(x)} \ dx$$ Any help would be greatly appreciated!
0
votes
1answer
19 views

Trigonometric Integral of variable function.

Let for any $n \in \mathbb Z$, define a function $f_n \text { on } [0,1]$ as follows: $$f_n(x) = \begin{cases}0 &\text{if} &x=0 \\ \sin ...
3
votes
1answer
65 views

Find $\int \cos^4(x)dx$

We have: $\int \cos^n x\ dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n}\int \cos^{n-2} x\ dx.$ Find $\int \cos^4x\ dx$ by using the formula twice What I have so far is: $\int \cos^4 x\ dx = ...
0
votes
1answer
30 views

How can you spot what trigonometric substitution to use in an integral?

So I seem to be coming across a lot of integrals requiring trigonometric substitutions. However, it's becoming tiring, because I have no idea how to spot what substitution should be used - i.e. ...
3
votes
1answer
55 views

Trigonometric Substitution

I am having trouble with this problem even though everything I did seemed right to me since we went over a similar one in my class. I used the method of setting up a triangle, my hypotenuse is ...
1
vote
1answer
26 views

Under what conditions are trigonometric integrals over a period zero?

Often times, while solving a physics problem, for example, an integral involving only sines and cosines (and constants), over a period, must be solved. In many cases this may prove difficult, so it's ...
0
votes
1answer
52 views

$\int t^2\cos(1-t^3)\,dt$ Explanation

I am trying to prep for a midterm. One of my practice problems is this $$\int t^2\cos(1-t^3)\,dt $$ Can someone show and explain how to do problems like these
2
votes
3answers
80 views

Evaluating $\int \frac{\sqrt{1-x^2}}{x^2} \operatorname d \! x$

I am trying to find $$\int \dfrac{\sqrt{1-x^2}}{x^2} \operatorname d \! x$$ I can get to $${\int \dfrac{\sin x}{\cos^2x}\operatorname d \! x}$$ I tried u substitution by making $u=\cos x$ and $du= ...
2
votes
1answer
21 views

Trigonometric Substitution

Question: Use the substitution x=3sin(t) to evaluate the integral of I started by making a right triangle and solving for sin(t) and cos(t). sin(t)=x/3 and cos(t)=(sqrt(9-x^2))/3 Then, I solved ...
3
votes
1answer
34 views

Sine substitution?

My book says the following: $$\int \frac{dx}{(16-x^2)^{3/2}}$$ $$x = 4\sin\theta$$ $$(16 - x^2)^{3/2} = (4^2\cos^2\theta)^{3/2}$$ $$=(4\cos\theta)^3$$ I don't understand the last step: Doesn't: ...
4
votes
1answer
39 views

Calculus 2 Trigonometric Integrals with odd exponents

I do not know how to go about taking the integral of this. I have tried to break it up so I can take the integral of sin^2(x) sin(x) cos^8(x) cos(x). But then this would require me use the reduction ...
4
votes
1answer
29 views

Calculus 2 Trigonometric Integrals

I have used the reduction formula for the integral of cos^n(x)dx but was unable to produce a correct answer. I think what throws me off is the (9x). Any helpful hints, tips, or other methods for ...
0
votes
1answer
40 views

Prove the integral of $\cot x$ is $-\ln|\csc x|+C$

I know how to prove it to be $\ln|\sin x|+C$, but I do not know the method to prove it this way. thanks
3
votes
1answer
74 views

Looking for the recurrence relation for certain trigonometric integrals

By assuming that: $$ \int_{\pi/4}^{\pi/2} \frac{\cos^4(x)}{\sin^5(x)}\,dx = k,$$ what does the integral $$ \int_{\pi/4}^{\pi/2} \frac{\cos^6(x)}{\sin^7(x)}\,dx$$ equal in terms of k? I have ...
0
votes
2answers
41 views

Integral of Form $(a^2 - x^2)^{1/2}$

Solve $\displaystyle \int \dfrac{1}{x^2(9-x^2)^{1/2}}\ \mathrm{d}x$ So far, I have used trigonometric substitution, so that $x = 3\sin y$, $x^2 = 9\sin^2 y$, and $\mathrm{d}x = 3\cos y\ \mathrm{d}y$. ...
3
votes
3answers
515 views

Integral of cos(5x)cos(3x)cos(4x)dx

The integral $\int_0^{\pi/8}\cos(5x)\cos(3x)\cos(4x) \,dx$ is equal to $k/24$. Find the constant $k$. So far, I assume that the best way to solve this question is to solve the integral and compare ...
0
votes
1answer
47 views

Integration of exponential trig functions

If $\cos^2x=[1+\cos(2x)]/2=(1/2)[1+\cos(2x)]$ Would I be wrong in assuming that $$\cos^2(3x+1)=\frac{1}{2}\left[1+\cos[2(3x+1)]\right]=\frac{1}{2}\cos(6x+2)+\frac{1}{2}?$$ I'm trying to take the ...
1
vote
3answers
81 views

Integral of $\int^\sqrt2_1\frac{1}{1+\sqrt{x^2 - 1}}dx$ by substitution?

In a maths question I have the question: $$\int^\sqrt2_1\frac{1}{1+\sqrt{x^2 - 1}}dx$$ by substitution? All other questions have been by trigonometric substitution so I assume that is how to solve. ...
2
votes
2answers
68 views

Did I solve this integral correctly? (trig substitution)

I'm having trouble with trig substitution. This is what I've done so far, but I'm not sure if I did everything right. This is the integral: $$\int \frac{x^2}{(1+x^2)^\frac{3}{2}}$$ and my ...
2
votes
1answer
77 views

Definite integral involving arctan and tan

I was solving a problem posed on Moldavian National Mathematical Olympiad for 12th grade in 2012. The question was the following: Problem. Let $f:\mathbb{R}\rightarrow\mathbb{R}$, such that ...
2
votes
2answers
93 views

Integral of complex questions?

$$\int_0^{\pi/4} \frac {\sin x + \cos x}{\sin^4x+\cos^2x}dx$$ $$\int e^x\cot x(\csc x-1)dx$$ These two integrals are impossible to find. If anyone knows how to integrate them please help me. I am ...
0
votes
2answers
48 views

Integrate Form $(a^2 +u^2)^{3/2}\ du$

I cannot solve that form... Here, is one of questions asked in that form Integrate Form $(x^2+ 4)^{3/2}dx$ The answer is $$ (1/4)(x^3+ 10x)(x^2+4)^{1/2}+ 6\ln\ \bigg(\frac{x + (x^2+ 4)^{1/2} ...
3
votes
1answer
40 views

Evaluate using a trigonometric substitution

The Integral is $$\int x^3\sqrt{9-x^2} dx$$ My method: $$x = 3\sin\theta$$ $$dx = 3\cos\theta d\theta$$ $$\sqrt{9-9\sin^2\theta} = 3\cos\theta$$ $$= \int 27\sin^3\theta (9\cos^2\theta) d\theta$$ ...
3
votes
5answers
204 views

How to evaluate this Trig integral?

I need to find the definite integral of $\int(1+x^2)^{-4}~dx$ from $0$ to $\infty$ . I rewrite this as $\dfrac{1}{(1+x^2)^4}$ . The $\dfrac{1}{1+x^2}$ part, from $0$ to $\infty$ , seems easy ...
0
votes
2answers
60 views

Are the sums equal to each other?

They are $2$ different results for the integral $$\int xe^{2x}\sin\left(\frac x3\right)\,dx$$ $\displaystyle\frac{-3}{1369}e^{2x}\left(3(35-74x)\sin\left(\frac x3\right)+(37x-36)\cos\left(\frac ...