Kirthi Raman
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 52 Funny identities 29 Which mathematicians have influenced you the most? 28 If $x$ and $y$ are rational numbers and $x^5+y^5=2x^2y^2,$ then $1-xy$ is a perfect square. 20 Numbers are too large to show $65^{64}+64^{65}$ is not a prime 18 Show by substitution that $\int_0^{\pi} \frac{x\sin x}{1+\cos^2 x} \,\mathrm dx = \frac{\pi}{2}\int_0^{\pi} \frac{\sin x}{1+\cos^2 x} \,\mathrm dx$

### Reputation (5,892)

 +20 Simplify $n^{\log\log n / \log n}$ +10 Integrating $\int \sin^n{x} \ dx$ +10 Show by substitution that $\int_0^{\pi} \frac{x\sin x}{1+\cos^2 x} \,\mathrm dx = \frac{\pi}{2}\int_0^{\pi} \frac{\sin x}{1+\cos^2 x} \,\mathrm dx$ +5 Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$

### Questions (25)

 24 Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$ 11 Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$ 6 Let $a,b$ be positive real numbers. Prove $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}} \geq \frac{2}{\sqrt{1+ab}}$ 6 Prove $1^a+2^a+\cdots+n^a < \frac{(n+1)^{(a+1)}-1}{a+1}$ for any $a >0$ and $n \in \mathbb{Z^+}$ 6 What is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$

### Tags (75)

 100 elementary-number-theory × 29 40 number-theory × 7 79 integration × 21 37 polynomials × 6 69 algebra-precalculus × 14 26 trigonometry × 11 64 calculus × 21 26 inequality × 9 40 sequences-and-series × 10 25 diophantine-equations × 6

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 Mathematics 5,892 rep 2152 Code Review 118 rep 3 Stack Overflow 101 rep 3