Can the definite integral $$\int_0^1\dfrac{\ln(1+x^2)}{1+x}dx$$ be evaluated using the technique of “differentiation under integral sign”. I don't want a complete solution, just the parameter would do.  PS: An alternative approach (preferably simple) would also be welcome as long as it doesn't involve contour integration.
I have an interest in contour integration. I am not that good at it, but I enjoy learning what I can about it.  Here is a version of a rational log integral rarely encountered.  $\displaystyle \int_{0}^{1}\frac{\ln(x^{2}+1)}{x+1}dx=\frac{3}{4}\ln^{2}(2)-\frac{{\pi}^{2}}{48}$  I can do this using real methods (via double integral and substitution). I can post my workings if anyone would be interested.  My question is, can this be evaluated using contour integration due to the limits being $[0,1]$ instead of $[0,\infty)$?.  Contours may not be the most efficient way to go about it, but what is the course of action when the limits are 0 to 1 instead of 0 to infinity?.
I'd like to update the label of an operator when it's called. But this :  def execute(self,context):     self.bl_label = "good night"   ... apparently doesn't work if the bl_label property has already been defined. Am I wrong ? Where does "wicked" get its /ɪd/ from? There are three ways I know to pronounce the -ed at the end of an adjective:   /t/ as in cracked. /d/ as in lined. /ɪd/ as in naked   I realise naked is a special case because, as , it comes for Old English nacod, so the suffix isn't added.   shows the pattern of pronunciation of -ed in all other cases. However it doesn't explain why  ends with /ɪd/ (or /id/ or /əd/ depending on the dialect).   that wicked comes from OE wicca, so the d wasn't already there.  Why do we pronounce wicked as /ˈwɪkɪd/ and not /ˈwɪkt/? Getting raw SQL query string from PDO prepared statements Is there a way to get the raw SQL string executed when calling PDOStatement::execute() on a prepared statement? For debugging purposes this would be extremely useful. I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more 'rigorous'. My understanding of a proof is that a proof is some explanation to convincing others that a statement is true. If some proof is not rigorous, does it mean that while the proof is acceptable for many but it is not for everybody? If so, the proof is not correct since it going to be rejected by some people and it is not a proof at all. Or, does rigorous mean it uses more advanced tools? If so, why not any alternative (but simpler) proof is also acceptable as rigorous (which is more 'valuable' as well since it's simpler!)?   Saying those limited insight of mine, my questions are:   $1-$ What does rigorous proof mean, in general definition?  $2-$ In order to understand your answers, as an example I am re-writing three of the proofs for the compactness of the torus. Which of the following proofs are most rigorous, 'normal' and least rigorous; and why so?    Theorem: The torus $T^2$ is compact.  First proof: The torus is homeomorphic to the product space $S^1 \times S^1$. The circle $S^1$ is compact. Therefore the torus is the product of two compact spaces, and thus it is compact.   Second proof: The torus $T$ is the subspace of $\mathbb{R}^3$ obtained by rotating a circle about the $z$-axis. The torus is closed since given any point in its complement, there is an open ball of sufficiently small radius centered at that point and disjoint from the torus. Also, the torus is bounded since it is contained in the ball of radius $4$ centered at the origin. Since the torus is closed and bounded in $\mathbb{R}^3$, it is compact.   Third proof: The torus is homeomorphic to a space obtained as a quotient  space of the square by identifying opposite edges. The square is a closed and  bounded subset of $\mathbb{R}^3$ and therefore is compact. Furthermore, a quotient map on the square is a continuous function, and the image of a compact space under a continuous function is compact. Therefore the torus is compact.   Thanks a lot for your guidance. Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer.  And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ diverges which also implies that the infinite series $\sum \limits_{k=0}^{\infty}a_k$ cannot be defined.  If I consider the infinite sum $1-1+1-1+1-1+\cdots$, then this statement is equivalent that I just defined the infinite series $\sum \limits_{k=0}^{\infty}a_k$, which is contradiction.  But, suppose that the sum $1-1+1-1+1-1+\cdots$ exists and let the value of the sum be $S$. Then, we can easily observe that $S=1-S$, therefore $S=1/2$. The supposition of this proposition already proved as false, but if I ignore the definition of infinite series, it holds.  Which one is right? Or are these propositions depends on what we define? Is the Macbook 12 inch's USB-C charger interchangeable with other USB-C chargers or devices? If there is only the Macbook 12 inch USB-C charger nearby, can it be used to charge other USB-C devices, such as any Android phones that also use USB-C? (will it be "too powerful" and damage other devices?)  Vice versa, if there is a Quick Charge 2.0 or 3.0 USB-C charger from other devices nearby, can it be used to charge the Macbook 12 inch?  The main concern is about whether it could damage the device or battery. How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$? Are the floor functions of $0.999\cdots$ and 1 equal?  It is true that $0.999\cdots=1$ but how does one justifies the integer part of $0.999\cdots$ being 1 , where it is not, or alternatively without using $0.999\cdots=1$ how can we show that  $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$ ? When and what cause Voldemort to look the way he does, vs how we've seen him as a teen? As the title says, when and what made Voldemort look the way he does (pale, snake-like etc) as opposed to how we've seen him in his earlier years as Tom Riddle? Is it just a side effect of his unnatural resurrection? Before he died attacking Harry as a baby did he look like a normal guy? How to use Mathematica functions in Python programs? How can I prove $n - 2\sqrt{n} = \Theta(n)$ Does anyone know the major differences between the Java and .Net garbage collectors? A web search has not revealed much, and it was a question that came up in a test. Database design for products with bundles of products How do you redirect to a page using the POST verb? When you call RedirectToAction within a controller, it automatically redirects using an HTTP GET.  How do I explicitly tell it to use an HTTP POST?  I have an action that accepts both GET and POST requests, and I want to be able to RedirectToAction using  POST and send it some values.  Like this:  this.RedirectToAction(     "actionname",     new RouteValueDictionary(new { someValue = 2, anotherValue = "text" }) );   I want the someValue and anotherValue values to be sent using an HTTP POST instead of a GET. Does anyone know how to do this? Are wormholes really a valid shortcut to distant points in the universe? Lately I encountered a few times references to wormholes as a way to reach distant places in the universe in a short time. I have to admit I don't have a proper knowledge of the mathematics behind wormholes (although I won't mind a complete GR explanation in the answer, if necessary). All I know is that wormholes are usually compared to something analogous to a tunnel through the planet Earth to reach distant places in less time. The problem is, if I want to reach the other side of the planet I have to travel 20.000 km on its surface and "only" 12.000 km through its center. While the latter is surely shorter, it's still more than 50% of the initial value and this is the best-case scenario. For shorter distances, the difference becomes more and more negligible.  Now, I know enough of GR to understand that this explanation could be a simplification so extreme that any comparison makes little to no sense, but I'm curious to understand if the same problems arise for wormholes too. I don't really care if wormholes exist, are traversable by spaceships and both the crew and the vessel can survive the travel undamaged, I know they are still a theoretical construct and probably there is more than one formulation for it. What I'm really interested in is:   How much shorter will the travel be? I'm interested in Earth proper time, because if the reduction is only in the spaceship proper time, the we can achieve the same result "simply" by traveling closer to the speed of light. Does the amount of reduction in time depend on the distance between the endpoints? If so, it seems unpractical to travel to near stars through wormholes, if the spacetime geometry of our galaxy doesn't help somehow (local curvature should be higher than global curvature). I compile my project with VIsual Studio 2005 Express Edition, and get this linking error. Can anyone suggest how to solve that? I have been running  over some C# code, and it keeps reporting that my using directives should be inside the namespace.  Is there a technical reason for putting the using directives inside instead of outside the namespace? Loop "Forgets" to Remove Some Items Show a,b,c congruences when $a^2+b^2=c^2$ i would like to know how to prove that if $a^2+b^2=c^2$ then   a) $3|a$ or $3|b$  b) $5|a$ or $5|b$ or $5|c$  c) $4|a$ or $4|b$   Also that $a^2+b^2+c^2+1$ can't be divisible by 8  Thank you very much! How to install the most common plugins? How to draw arrows between parts of an equation to show the Math Distributive Property (Multiplication)? There is a part of my paper where I use the distributive property and it might not be clear to the reader to see the operation result (I'm using in proposional logic for a proof). The logic equation (part of) is: $$ (l_{1}) \vee \big( (p \vee q) \wedge (\neg p \vee q) \wedge (p \vee \neg q) \wedge (\neg p \vee \neg q)\big) $$ $$ (l_{1} \vee p \vee q) \wedge (l_{1} \vee \neg p \vee q) \wedge (l_{1} \vee p \vee \neg q) \wedge (l_{1} \vee \neg p \vee \neg q) $$  So I was wondering if there is a way to represent this operation like a 'shower' (see another example in pic ). Maybe there is some hidden magic mathmode command or pkg for this. At last we can also rely on tikz.