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$Proving a reduction formula for the antiderivative of $\cos^n(x)$ [duplicate]$

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Proving a reduction formula for the antiderivative of $\cos^n(x)$ [duplicate]

I want to show that for all $n\ge 2$, it holds that $$ \int \cos^n x\ dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n}\int \cos^{n-2} x\ dx. $$ I'...

Published: Sep 05, 2026

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How far is the list of known primes known to be complete?

So there is always the search for the next "biggest known prime number". The last result that came out of GIMPS was $2^{74\,207\,281} - 1$, with over twen...

Published: Sep 05, 2026

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How to prove the Closed map lemma

The closed map lemma says that if $f : X \to Y$ is a continuous function, $X$ is compact and $Y$ is Hausdorff, then $f$ is a closed map. How can I prove t...

Published: Sep 05, 2026

$Quaternions: Why is the angle $\frac{\theta}{2}$? [duplicate]$

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Quaternions: Why is the angle $\frac{\theta}{2}$? [duplicate]

The equation for creating a quaternion from an axis-angle representation is $$x'= x \sin\left(\frac \theta 2\right)$$ $$y' = y \sin\left(\frac \...

Published: Sep 05, 2026

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How do I find the equation of a parabola given the max, and two points?

I was given the points (2, -1) and (10,-1) and also a max of 4. How would I go about finding the equation of the parabola given this info?

Published: Sep 05, 2026

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Prove the sum to product formulas with complex numbers

First of all, I don't know if it's possible but I'm just wondering about it. I'm pretty standard when it comes to complex numbers know...

Published: Sep 05, 2026

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The relationship between spectral decomposition / eigendecomposition and projection operators

I am trying to clarify the relationship between the spectral decomposition / eigendecomposition of a matrix and projection operators. I understand that th...

Published: Sep 05, 2026

general

A detail in the proof of Auslander-Buchsbaum Theorem

I'm trying to understand the proof of a theorem (Auslander-Buchsbaum) which says that given a local ring $(R,m)$, where $m$ is the maximal ideal, and...

Published: Sep 05, 2026

general

Find the volume $V$ of the described solid $S$.

Find the volume $V$ of the described solid $S$. The base of $S$ is the triangular region with vertices $(0, 0), (4,0)$, and $(0, 4)$.Cross-sections perpen...

Published: Sep 05, 2026

$Antiderivative of $\sin(x-\pi/3)$$

general

Antiderivative of $\sin(x-\pi/3)$

I used $u$-substitution to break down the problem to be $-\cos(u)$ and then got my final answer to be $-\cos(x-\pi/3)$ (re-insertion for my $u$) and it is...

Published: Sep 05, 2026

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