Intereting Posts

Are there many more irrational numbers than rational?
A simple explanation of eigenvectors and eigenvalues with 'big picture' ideas of why on earth they matter
Likelihood Function for the Uniform Density $(\theta, \theta+1)$
Determining the generators of $I(X)$
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $
Calculate Laurent series for $1/ \sin(z)$
How to generate a random number between 1 and 10 with a six-sided die?
Countable axiom of choice: why you can't prove it from just ZF
Closure of the span in a Banach space
Derangements with repetitive numbers
Evaluation of $\int\frac{(1+x^2)(2+x^2)}{(x\cos x+\sin x)^4}dx$
Singular $\simeq$ Cellular homology?
specifying the joint distribution as a proof technique
Inverse of a matrix is expressible as a polynomial?
Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q \implies \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $

$$\int x\sqrt{x+x^2}\,dx = ?$$

Can you solve this problem? I have tried this after a challenge. 😊

- Equality with Euler–Mascheroni constant
- Calculate $\int_0^\infty {\frac{x}{{\left( {x + 1} \right)\sqrt {4{x^4} + 8{x^3} + 12{x^2} + 8x + 1} }}dx}$
- show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$
- A common term for $a_n=\begin{cases} 2a_{n-1} & \text{if } n\ \text{ is even, }\\ 2a_{n-1}+1 & \text{if } n\ \text{ is odd. } \end{cases}$
- Can we teach calculus without reals?
- Convergence\Divergence of $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$

- Probabilistic proof of existence of an integer
- Proof for power functions
- Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$
- Proving the series of partial sums of $\sin (in)$ is bounded?
- Series that converge to $\pi$ quickly
- Horse and snail problem.
- Left Inverse: An Analysis on Injectivity
- A inverse Trigonometric multiple Integrals
- Ramanujan log-trigonometric integrals
- Integrating the following $\int \sqrt{\tan x+1}\,dx$

**Hint:**

Set $x^2+x=u$, we have

$$\int x\sqrt{x+x^2}\,dx =\frac 12 \int (2x+1)\sqrt{x+x^2}\,dx-\frac 12 \int \sqrt{-\frac 14+\left(x+\frac 12\right)^2}\,dx$$

Let $x+\frac12=\frac12\cosh(u)$, then $\sqrt{x+x^2}=\frac12\sinh(u)$ and $\mathrm{d}x=\frac12\sinh(u)\,\mathrm{d}u$. Furthermore, $\sinh^2(u)=\frac{\cosh(2u)-1}2$

$$

\begin{align}

\int x\sqrt{x+x^2}\,\mathrm{d}x

&=\frac18\int(\cosh(u)-1)\sinh^2(u)\,\mathrm{d}u\\

&=\frac1{24}\sinh^3(u)-\frac1{32}\sinh(2u)+\frac1{16}u+C\\

&=\frac1{24}\sinh^3(u)-\frac1{16}\sinh(u)\cosh(u)+\frac1{16}u+C\\

&=\frac13\sqrt{x+x^2}^{\,3}-\frac18(2x+1)\sqrt{x+x^2}+\frac1{16}\cosh^{-1}(2x+1)+C

\end{align}

$$

Hint: You can do that easily using Euler’s substitution.

- Maximum area of a rectangle inscribed in a triangle is $1/2$ the area of triangle
- Describe all $p^{n}$ (in terms of congruence conditions of $p$ and $n$) for which $x^{2}+1$ irreducible over $\mathbb{F}_{p^{n}}$.
- The $n$-th derivative of the reciprocal of a function and a binomial identity
- If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
- Prove that $\phi(n) \geq \sqrt{n}/2$
- homeomorphism of topological spaces is an equivalence relation ?
- Proving that a map is a weak homotopy equivalence
- Universal Definition for Pullback
- Bijection between $\mathbb{R}$ and $\mathbb{R}/\mathbb{Q}$
- Can any torsion free abelian group be embedded in a direct sum of copies of $\mathbb Q$?
- Proving $(p \to (q \to r)) \to ((p \to q) \to (p \to r))$
- Non-backtracking closed walks and the Ihara zeta function (Updated with partial attempt)
- Find the sum of the series $\sum^{\infty}_{n=1} \frac{1}{(n+1)(n+2)(n+3) \cdots (n+k)}$
- Integration double angle
- What is the support of a localised module?