Why is the integral of $0.5/x = 1/2 \ln x$ and not $1/2 \ln 2x$?
Two different solutions to integral
Find the value of $x,y$ for the equation $12x+5y=7$ using number theory . I tried the solution using other ways like:  $12x+5y=7$  $12x=7-5y$  $x=7-5y/12$  putting the value of $x$  $12(7-5y/12)+5y=7$  both $12$ will cancel out  $7-5y+5y=7$ here comes the problem please help me ? Does Windows Defender update if Windows Update is set to manual? Is there a word to describe words that can be spelled different ways? Is there a word in English to describe words that can be spelled more than one way?  For example:   colour and color favourite and favorite   However, this shouldn't be limited to words spelled differently in different dialects, though I'm not sure what other case there would be. What is EV, when used as an absolute measurement? Seatpost Overextension I've been riding with my very short seat post extended well past its maximum extension line for quite a while now. Looking at the seat post itself I don't see any signs of deformation or damage other than some very minor surface rust.   Could my bike frame have been damaged? Could the lever force have damaged the frame? I can post some pictures of the frame if needed. My bike is all aluminum. Why I can't read a read only file? I’m able to copy-paste from any application to any application. I’m able to copy from a comment or from a question or from an answer. I’m able to paste to a question or answer. However, if I attempt to paste into a comment, nothing happens. I am able to subsequently paste into another application, confirming that the text is actually on the clipboard.   However, copying SE links will paste them successfully, but all other text - including other links - still fails. This leads me to believe that it’s a SE issue and not an Apple one.   This has been an issue since 11.0 beta and persists through iOS 13 release. How to prove the following expression does not depend on x? I'm not sure how to approach this question, but I know it has something to do with taking the derivative of this. Can anyone please help me out?  Let x > 0. Prove that the value of the following expression does not depend on x: $$\int_{0}^{x}\frac{1}{1+t^4} dt + \frac{1}{3}\int_{0}^\frac{1}{x^3} \frac{1}{1+t^\frac{4}{3}} dt$$ How to roll back Ubuntu to a previous version? I need to install a new TeX/LaTeX/ConTeXt package on my computer.  I'm running Ubuntu/Fedora/Slackware/Some-Version-Of Linux.  What are the possible ways to do this?    Note:&nbsp;&nbsp; We get a number of questions that essentially reduce to this one; it was felt a good idea to have one question-and-answer that can cover all the standard methods.  See  for the discussion.  The question-and-answer are CW so that more information can be easily added.  If you have information to add, please do so.  If you are unsure, leave a comment here or (preferably) in .  Comments on the question and answer will be periodically removed to ensure that the information is clear. I would like to enable the root account on my Ubuntu installation.  I realize the security implications of this. Adding values to a C# array permission denied BASH Differentiation - simple case In the book calculus made easy, page 22 the case of the negative power for   $y=x^{-2}$    $$\begin{align}  y+dy &amp; =(x+dx)^{-2}\tag{1}\\ \\ &amp; = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} \end{align}$$  How do they get from $(1)$ to $(2)$?  And why isn't the binomial function applied to the second line but applied on the third line (the line that comes after line $(2))$? Are the XBLA and PC versions of Minecraft compatible? Here the lower box dimension and the upper box dimension are exactly what Falconer talks about in his book "Fractal Geometry". I already know that the upper box dimension is finitely stable since $N_\delta(A\cup B)\le N_\delta (A)+N_\delta(B)$, where $N_\delta(A)$ is defined as usual, and Falconer gives an example to show the box dimension may not be countably stable. (We know that, however, the Hausdorff dimension is countably stable).  But now I want to find the example to show the lower box dimension may not even be finitely stable.It seems that we can use von Koch curve to illustrate this, but what is the explicit explanation for this? Since the lower box dimension is monotonic, we need to find set A and B and show that   $\underline{\dim}_B(A\cup B)&gt;\max\{\underline{\dim}_B(A),\underline{\dim}_B(B)\}$. What are the limits of house-elf apparition? I know Wizards can't apparate where they want on the Earth, except if they are very powerful but what about a house-elf ? Their magic is different so I was wondering if they can apparate wherever they want or if they have a limit too. If $z_n \to z$ then $(1+z_n/n)^n \to e^z$ We are dealing with $z \in \mathbb{C}$. I know that $$ \left(1+ \frac{z}{n} \right)^n \to e^{z} $$ as $n \to \infty$. So intuitively if $z_n \to z$ then we should have $$ \left(1+ \frac{z_n}{n} \right)^n \to e^{z}. $$ If $z_n \in \mathbb{R}$ I would be happy writing $$ \exp \left(n \log\left(1+ \frac{z_n}{n} \right) \right) = \exp \left(z_n \frac{\log\left(1+ \frac{z_n}{n} \right)-\log(1+0)}{\frac{z_n}{n}-0} \right) \to \exp(z \cdot 1) $$ where here we are using the definition of derivative. But if $z \in \mathbb{C}$ the logarithm is multivalued and I'm not sure that the same calculation is allowed. Is there a different way to show this for complex $z$? What is the inverse of CoefficientList? what is the difference between IBE and ABE schemes Can somebody explain in simple words. real life example light help and limitations of these schemes etc.