# Tagged Questions

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### Integral of $\cos(\cos x)$ over $[0,2\pi]$

How to compute the following integral? $$\mathcal{J}_2=\int_{0}^{2\pi}\cos(\cos t)\,dt$$ I'm trying to compute this integral, but I have no idea of how to do it, can someone help me?
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### How to find the approximate basic frequency or GCD of a list of numbers?

I could't actually summarize the question in the title, so I'll explain my situation. I want to tell the integer numbers which act as the best approximate basic frequencies of a list of real numbers: ...
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### Scratching my head whith a problem of infinity.

The first equation in a) gives a sum of 1 and the second equation starts with a sum equal to $\pi$ or $180$ in radian and degree mode.By removing the sign from $\sqrt x$ in b) the value of y is still ...
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### Nonsensical result in the midst of calculating an integral via substitution.

I was just calculating an integral via a trigonometric substitution and ended up with $\color{red}{ \text{something pretty nonsensical} }$ but $\color{blue}{ \text{reversing the substitution} }$ ...
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### Evaluate$\int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$

Find the value of the integral $\int_0^{\frac{\pi}{2}} \ln(1+\cos x)$ I tried putting $1+ \cos x = 2 \cos^2 \frac{x}{2}$, but am unable to proceed further. I think the following integral can be ...
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### Is $f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$ strictly positive?

I would like to have an advice for this exercise. Let $x\in[0,\pi]$ For which values of $x$ this function $$f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$$ is strictly positive ? I tried to ...
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### Implicit differentiation of trig functions

I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation: $$\cos\cos(x^3y^2) - x \cot y = -2y$$ I figured that the calculation requires the chain ...
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### Why $\cos^2 x-\sin^2 x = \cos 2x\;?$

I was hoping someone could explain how $\cos^2 x-\sin^2 x = \cos 2x$ After using the product rule to differentiate $\sin x \cdot \cos x$ I get the answer $\cos^2 x - \sin ^2 x$ I've come across this ...
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### Differentiate $(x-1)^2 \sin x$ where $x$ is in radians

How would I differentiate, simplify and then find $f'(\pi/2)$: $$f(x)=(x-1)^2 \sin x$$ I'm not sure how to differentiate $\sin x$ to then use it later to find an answer, any help would be much ...
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### How can I find $\lim_{x \to 0}\frac{\tan(3x)}{\sin(8x)}$ without L'Hospital's Rule

Is there a way to solve $\lim_{x \to 0}\frac{\tan(3x)}{\sin(8x)}$ without using the trig identity $\tan(3x)=\frac{3\tan(x)-\tan^3(x)}{1-\tan^2(x)}$. I want to know because I had to look up this trig ...
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### Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
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### How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
### Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]
As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.