Hello I am solving an integral with a natural logarithm that has a parameter. Let say $I(a)=\int_0^\pi\ln(1-2a\cos(x)+a^2)dx$ Then for differentiation under integral sign and that yields $I'(a)= \int_0^\pi\frac{2a-2\cos(x)dx}{ 1-2a\cos(x)+a^2}$.  I chose take $aI'(a)$ and then I have That $I(a)= \pi\ln(a^2)+a\pi$  My questions:    is there any wrong over result? where I shoud to take considerations about $a$? magnitude. what is the integral problem about this exercise?   Thanks, any help, any correction.
For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$.   My attempt: Let  $$\begin{align*} f_n(x) &amp;= \frac{\ln\left(1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2\right)-\ln\left(1-2a\cos x+a^2\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(\displaystyle\frac{1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2}{1-2a\cos x+a^2}\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(1+\dfrac{1}{n}\left(\displaystyle\frac{2a-2\cos x+\frac{1}{n}}{1-2a\cos x+a^2}\right)\right)}{\frac{1}{n}}. \end{align*}$$ Now it is easy to see that $f_n(x) \to \frac{2a-2\cos x}{1-2a\cos x+a^2}$ as $n \to \infty$. $|f_n(x)|\le \frac{2a+2}{(1-a)^2}$ RHS is integrable so $\lim_{n\to\infty}\int_0^\pi f_n(x)dx = \int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2} dx=I'(a)$. But  $$\int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2}=\int_0^\pi\left(1-\frac{(1-a)^2}{1-2a\cos x+a^2}\right)dx.$$ Consider $$\int_0^\pi\frac{dx}{1-2a\cos x+a^2}=\int_0^\infty\frac{\frac{dy}{1+t^2}}{1-2a\frac{1-t^2}{1+t^2}+a^2}=\int_0^\infty\frac{dt}{1+t^2-2a(1-t^2)+a^2(1+t^2)}=\int_0^\infty\frac{dt}{(1-a)^2+\left((1+a)t\right)^2}\stackrel{(*)}{=}\frac{1}{(1-a)^2}\int_0^\infty\frac{dt}{1+\left(\frac{1+a}{1-a}t\right)^2}=\frac{1}{(1-a)(1+a)}\int_0^\infty\frac{du}{1+u^2}=\frac{1}{(1-a)(1+a)}\frac{\pi}{2}.$$ So $$I'(a)=\frac{\pi}{2}\left(2-\frac{1-a}{1+a}\right)\Rightarrow I(a)=\frac{\pi}{2}\left(3a-2\ln\left(a+1\right)\right).$$  It looks too easy, is there any crucial lack?  $(*)$ — we have to check $a=1$ here by hand and actually consider $[0,1), (1,\infty)$ but result on these two intervals may differ only by constant - it may be important but in my opinion not crucial for this proof.
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Transvection matrices generate $ \operatorname{SL}_n(\mathbb{R}) $ Pointwise limit of continuous functions not Riemann integrable UK Visa Refusal: Provenance of funds/parking For my MAC laptop, I only have 250GB and out of this 250GB MAC is taking up 129GB in the other section. I looked everywhere to the best of my knowledge. Please note my home directories is only 45GB. Can you somehow please help me? I want to add additional items to the data returned as . So for every function/query that returns a WP_Post object I want my additional data added.  Example of returned result:  WP_Post (object) =&gt; [     // Default data returned by WP_Post     ID                  =&gt; int,     post_author         =&gt; string,     post_name           =&gt; string,     post_type           =&gt; string,     post_title          =&gt; string,     post_date           =&gt; string,     post_date_gmt       =&gt; string,     post_content        =&gt; string,     post_excerpt        =&gt; string,     post_status         =&gt; string,     comment_status      =&gt; string,     ping_status         =&gt; string,     post_password       =&gt; string,     post_parent         =&gt; int,     post_modified       =&gt; string,     post_modified_gmt   =&gt; string,     comment_count       =&gt; string,     menu_order          =&gt; string,      // Additional data I want to add     extra_data_1        =&gt; array,     more_data_key       =&gt; string,     another_added       =&gt; string ]   For example when the functions  or  are run they will return the WP_Post object along with my additional data.  I've tried finding the appropriate  but this has been unsuccessful.  I am hoping I can do something like:  // This is concept code add_action('pre_wp_post_return', function($data) {     $data-&gt;extra_data_1     = get_post_meta($data-&gt;ID, 'extra_data');     $data-&gt;more_data_key    = get_post_meta($data-&gt;ID, 'more_data', true);     $data-&gt;another_added    = get_post_meta($data-&gt;ID, 'another_data', true);      return $data; });   My reasoning is I am having to build a custom API for WP that uses a range of different core functions which return WP_Post objects. Being able to add my additional data in one place would prevent me from duplicating code.  I hope this is clear enough. Any help would be great! Prove $\chi(G)\chi(\bar{G}) \geq n$ for chromatic number of graph and its complement This is a  about IPv6 Subnetting.      Related:               I know a lot about , and as I prepare to (deploy|work on) an IPv6 network I need to know how much of this knowledge is transferable and what I still need to learn. IPv6 seems at first glance to be much more complex than IPv4. So I would like to know:   IPv6 is 128 bits, so why is /64 the smallest recommended subnet for hosts? Related to this:   Why is it recommended to use /127 for point to point links between routers, and why was it recommended against in the past? Should I change existing router links to use /127? Why would virtual machines be provisioned with less than a /64 worth of addresses? Are there other situations in which I would use a subnet smaller than /64?  Can I map directly from IPv4 subnets to IPv6 subnets? For instance, does an IPv4 /24 correspond directly to an IPv6 /56 or /120? My interfaces have several IPv6 addresses. Must the subnet be the same for all of them? Why do I sometimes see a % rather than a / in an IPv6 address and what does it mean? Am I wasting too many subnets? Aren't we just going to run out again? In what other major ways is IPv6 subnetting different from IPv4 subnetting? I need to customize my references in the following style using biblatexs Alpha style having the columns Label, Author, and Title and Other Information.    Till now with your help I could already change the square brackets to slashes.   My code so far is:   \documentclass[12pt]{scrartcl}  .  .   \usepackage[% natbib        = true, backend       = bibtex8, %bibtex, biber style         = alphabetic, %citestyle     = alphabetic, maxcitenames  = 2, mincitenames  = 1, %bibstyle      = alphabetic, %authoryear, %draft, %, sorting       = nyvt, maxbibnames   = 6, minbibnames   = 6, language      = ngerman, date          = long, backref       = false, backrefstyle  = none,  % none, three, two, two+, three+, all+ firstinits=true  ]{biblatex}    \usepackage[babel,german=quotes]{csquotes}    \bibliography{input/literatur}      \newrobustcmd{\mkbibslashes}[1]{/#1/}      \DeclareFieldFormat{labelalphawidth}{\mkbibslashes{#1}}    \DeclareFieldFormat{shorthandwidth}{\mkbibslashes{#1}}     \DeclareCiteCommand{\cite}[\mkbibslashes]    {\usebibmacro{prenote}}    {\usebibmacro{citeindex}% \usebibmacro{cite}}    {\multicitedelim}    {\usebibmacro{postnote}}    \renewcaptionname{ngerman}{\refname}{Literaturverzeichnis}   .   .   .   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Norms Induced by Inner Products and the Parallelogram Law If i entered "name" in my text box while it saved it must be saved like "Name"   Capital letters in the first letter of Alphabet  It must not show any error messages it must directly save Fundamental Theorem of Calculus. Fundamental Theorem of Calculus says  Let $f$ be an integrable function on $[a,b].$ For $x$ in $[a,b],$  let $F(x)=\int_{a}^{x}f(t)dt.$ Then $F$ is continuous on $[a,b].$ If $f$ is continuous at $x_{0}$ in $(a,b),$ then $F$ is differentiable at $x_{0}$ and $F^{'}(x_{0})=f(x_{0}).$  Now my question is why we are saying that this is "Fundamental Theorem of whole Calculus".  It is clear that it connects intergal and differential calculus. But i am not getting the exact concept, why this is so important as its name is fundamental theorem of calculus. Why we says that it is fundamental theorem of calculus?  Please suggest me. Thanks in advance. 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