10 Evaluate $\displaystyle\int\limits_{-2}^{2}\frac{x^{4}}{1+6^{x}}dx$ 8 Show that $M[x] \cong A[x] \otimes_{A} M.$ 8 Let $f: \mathbb R \longrightarrow \mathbb R$ be $\mathcal C^\infty$ with $f'(x) = f(1-x)$ for all $x$. Given $f(0)=1$. Find $f(1)$. 7 Group structure of this quotient group 6 Convergence and limit of $a_1 - a_2 +a_3-a_4 + \cdots$, where $a_n=\frac{1+\frac{1}{2}+\frac{1}{3} + \cdots +\frac{1}{n}}{n}$

### Reputation (8,389)

 +10 Proof of $k$th derivatives always being integers +13 Subfields extension that generates a field implies inequality +10 Can we construct $72^{\circ}$ with a compass and ruler? +10 $(\mathbb{Q},+)$ and $(\mathbb{Q} \times \mathbb{Q},+)$ are not isomorphic as groups

### Questions (14)

 18 Is every group the unit group of some ring? 6 Why is a manifold called a manifold? 5 On an Infinite Series 4 The closure of meromorphic functions under composition 3 $n$ as an integer and a real number

### Tags (188)

 98 calculus × 44 34 group-theory × 21 68 abstract-algebra × 40 32 ring-theory × 17 64 sequences-and-series × 32 32 algebra-precalculus × 16 60 integration × 22 31 definite-integrals × 10 53 real-analysis × 28 30 elementary-number-theory × 21

### Bookmarks (28)

 80 Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$ 29 Divergence of the Derivative of the Prime Counting Function 29 Closed form of $\frac{e^{-\frac{\pi}{5}}}{1+\frac{e^{-\pi}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-3\pi}}{1+\ddots}}}}$ 27 How to properly use GAGA correspondence 24 A nice formula for the Thue–Morse sequence