Intereting Posts

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Riemann integrability of continuous function defined on closed interval
How are $G$-modules and linear group actions different
Homemorphism from $S^n$ to $S^n$
Arithmetic-geometric Mean
differential system on the torus

I am asked to integrate by parts $\int \ln(x) dx$. But I’m at a loss isn’t there supposed to be two functions in the integral for you to be able to integrate by parts?

- What is integration by parts, really?
- Probability of two people meeting during a certain time.
- Useful techniques of experimental mathematics (reference request)
- If $\sum\limits_{k=1}^{\infty}a_k=S $, then $ a_4+a_3+a_2+a_1+a_8+a_7+a_6+a_5+\dots=?$
- Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$
- Prove that $\lim \limits_{n\to\infty}\frac{n}{n^2+1} = 0$ from the definition
- The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$)
- Solving $\lim\limits_{x\to0} \frac{x - \sin(x)}{x^2}$ without L'Hospital's Rule.
- How do I solve this improper integral: $\int_{-\infty}^\infty e^{-x^2-x}dx$?
- How to find the value of $\sqrt{1\sqrt{2\sqrt{3 \cdots}}}$?

**Hint**: Write $\log(x)$ as $1 \cdot \log(x)$ and use integration by parts.

$$\int \ln x \, dx= \int 1\cdot \ln x\, dx= x\ln x- \int x\cdot \frac{1}{x}\, dx=x\ln x-\int \,dx = x\ln x-x+C,$$

where $C$ is a constant.

$$

\int \ln x\,dx = \underbrace{\int u\,dx = ux – \int x\,du}_{\text{integration by parts}} = x\ln x – \int x\,\frac{dx}{x}.

$$

Now cancel the $x$ from the numerator and denominator and go from there.

(I’ve seen probably at least a couple of dozen students fail to figure out that a cancelation can be done there. They wonder about such things as whether one should integrate the $x$ and the $dx/x$ separately and then multiply.)

The arctangent function is done the same way:

$$

\begin{align}

\int\arctan x\,dx & = \underbrace{\int u\,dx = ux – \int x\,du}_{\text{integration by parts}} = x\arctan x – \int x\, \frac{dx}{1+x^2} \\[8pt]

& = x\arctan x – \int\frac{1}{1+x^2} \cdot\frac12\cdot \Big(2x\,dx\Big) \\[8pt]

& = x\arctan x – \frac12\int\frac1w\,dw \\[8pt]

& = x\arctan x-\frac12\ln |w| + C \\[8pt]

& = x\arctan x – \frac12\ln(1+x^2)+C.

\end{align}

$$

Just a good point: Wherever you have $$\int p(x)\ln(x)dx$$ in which $p(x)$ is an integrable function; you can use the integration by parts as follows: $$u=\ln(x),~~dv=p(x)dx$$ such that $$\int udv=uv-\int vdu$$

- Needing help picturing the group idea.
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- Closed form for ${\large\int}_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$
- Identity with Catalan numbers
- Bases having countable subfamilies which are bases in second countable space
- Recurrence with varying coefficient
- Squarefree binomial coefficients.
- Density of $\mathcal{C}_c(A\times B)$ in $L^p(A, L^q(B))$
- Sum of digits and product of digits is equal (3 digit number)
- To find the nilpotent elements of $\Bbb Z_n$ and also the number of nilpotent elements of $\Bbb Z_n$.
- Where are the geometric figures in those “advanced geometry” textbooks?
- Diophantine Equations : Solving $a^2+ b^2=2c^2$
- Prove that $\tan A + \tan B + \tan C = \tan A\tan B\tan C,$ $A+B+C = 180^\circ$
- Where does the word “torsion” in algebra come from?
- Prove that any set of 2015 numbers has a subset who's sum is divisible by 2015