I have the integral:      $$\int_{-\infty}^\infty e^{-x^2} \, dx$$   And I’d like to solve it using differentiation under the integral sign. I understand that I must convert $e^{-x^2}$ to $e^{-x^2}g(x,t)$, where $g(x,t)$ is just some term involving a new parameter $t$. I tried multiplying by $e^{-2xt}$, which doesn’t really seem very logical, however, I just do not know what to put. How do I find the appropriate term to make the whole process work?
I stumbled upon  on last weekend, it introduces an integral trick that exploits . On its last page, the author, Mr. Anonymous, left several exercises without any hints, one of them is to evaluate the Gaussian integral $$ \int^\infty_0 e^{-x^2} \,dx= \frac{\sqrt{\pi}}{2} $$ using this parametrization trick. I had been evaluating it through trial and error using different paramatrizations, but no luck so far.     Here are what I have tried so far:   A first instinct would be do something like:$$ I(b) = \int^\infty_0 e^{-f(b)x^2}\,dx $$ for some permissible function $f(\cdot)$, differentiating it will lead to a simple solvable ode: $$ \frac{I'(b)}{I(b)} = -\frac{f'(b)}{2f(b)} $$ which gives: $$ I(b) = \frac{C}{\sqrt{f(b)}}. $$ However, finding this constant $C$ basically is equivalent to evaluating the original integral, we are stuck here without leaving this parametrization trick framework. A second try involves an exercise on the same page: $$ I(b) = \int^\infty_0 e^{-\frac{b^2}{x^2}-x^2}dx. $$ Taking derivative and rescaling the integral using change of variable we have: $$ I'(b) = -2I(b). $$ This gives us another impossible to solve constant $C$ in: $$ I(b) = C e^{-2b} $$ without leaving this framework yet again. The third try is trying modify Américo Tavares's answer in this : $$ I(b) = \int^\infty_0 be^{-b^2x^2}\,dx. $$ It is easy to show that: $$ I'(b) = \int^\infty_0 e^{-b^2x^2}\,dx - \int^\infty_0 2b^2 x^2 e^{-b^2x^2}\,dx = 0 $$ by an integration by parts identity: $$ \int^\infty_0 x^2 e^{- c x^2}\,dx = \frac{1}{2c}\int^\infty_0 e^{- c x^2}\,dx . $$ Then $I(b) = C$, ouch, stuck again at this constant.     Notice in that  question, Bryan Yocks's answer is somewhat similar to the idea of parametrization, however he has to introduce another parametric integration to produce a definite integral leading to $\arctan$.  Is there such a one shot parametrization trick solution like the author Anonymous claimed to be "creative parameterizations and a dose of differentiation under the integral"?
Notepad doesn't open files created by Gedit? If $G$ has no proper subgroup, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.     I know that since $G $is a group with no proper subgroups, $g \in G$ is not just the identity. I don't know where to go from there. Server vs single-player world effects Are there any differences with regards to chunk generation or other settings in single-player vs server mode of equivalent versions?  I have a map I'd like to host on a private server for my friend to play with me. However, when I'm on the road without ready internet, I'd like to play it single-player as well.  Will this cause any issues, especially with chunk generation, drops, items, etc (i.e., getting a different chunk generated in server-mode than single-player would have created), or do all server-client version pairs (I'm using 1.6.2) use the same game code?  The main point is that I don't want gameplay to change at all, between single-player and server mode. Switch Stack Exchange site's interface to the beta theme How can I convert String to Int? I'm having some trouble proving the following statement using mathematical induction:   $$\frac{1}{2}n^{\frac{3}{2}} \leq \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} + ... + \sqrt{n}      ,\text{ (for all sufficiently large n)}$$   I'm sort of confused because $n^{\frac{3}{2}}$ = $n  n^{\frac{1}{2}}$ which means that there are n of the largest possible root values on the left hand side while the right hand side has values like $\sqrt{1}$, $\sqrt{2}$, etc. As n grows sufficiently large wouldn't the left hand side outgrow the right hand side no matter what the constant multiple is? If I'm thinking about this correctly just let me know, if not, any help would be appreciated. I've always wondered this - why can't you declare variables after a case label in a switch statement?  In C++ you can declare variables pretty much anywhere (and declaring them close to first use is obviously a good thing) but the following still won't work:  switch (val)   {   case VAL:     // This won't work   int newVal = 42;     break; case ANOTHER_VAL:     ...   break; }     The above gives me the following error (MSC):     initialization of 'newVal' is skipped by 'case' label   This seems to be a limitation in other languages too.  Why is this such a problem? When I want to register an account with Gmail now, it says "Google will send a text message containing a verification code to your mobile phone."  I don't want to give Google my mobile phone number!!  How do I get around this? Why are drivers and firmwares almost always written in C or ASM and not C++? How do you remove the axis multiplier? A chart with data in the range of 200,000 and small difference between max and min values, will be assigned a 10^5 multiplier to the entire axis. This results in the ticks becoming meaningless, since all become 2.   Example:-  \begin{tikzpicture} \begin{axis}   \addplot coordinates {   (100,200001)   (200,200003)   (300,200005)   };     \end{axis} \end{tikzpicture}   When this is plotted the y-axis has 5 ticks, all of which are 2. The y-axis has a multiplier of 10^5. This has effectively removed all information from the graph.  I may use the below to set the explicit value of the tick values but this does not implement change in the display "style" of the axis ticks.  ytick = {200000, 200005, 200010}   I have also tried increasing the precision of the tick label to 5. However this only results in the graph showing labels such as 2.00000, 2.00005, 2.00010 with a 10^5 axis multiplier.  yticklabel style={precision = 5,}   My question is "How do I remove/change this 10^5 multiplier, such that the y-axis ticks are displayed in the form 200000, 200005, 200010. Escaping quotes in zsh alias Following on from  about stripping newlines out of text, I want to turn this into a zsh alias as follows:  alias striplines=' awk " /^$/ {print \"\n\"; } /./ {printf( \" %s \",$0);}"'   I've tried escaping the quotes inside the awk script, but I'm getting this error:  awk: (FILENAME=bspsrobustness FNR=1) fatal: division by zero attempted   (The file is called bspsrobustness)  Is there a way to do what I want? I suppose I could turn this into an awk script rather than a zsh alias, is that my best option? Hello [Comma?] John, How do i prove invertibility of this linear transformation Let $T$ be a linear transformation from a vector space $V$  over reals into $V$ such that $T-T^2=I$. Show that $T$ is invertible Suppose a book consists of three sections:     The book is/was divided into three sections.   I am thinking the past tense option "was" should be used, because the act of dividing the book into three sections happened at the time of writing. Is my analysis good? Turning off "Update page numbers only" in Word When I update the fields document-wide (Ctrl + A, then F9) in a Word document that has a table of contents, a dialog appears asking if I want to update page numbers only, or update the entire table. I never want to update page numbers only if the document headings have changed, nor will I ever.  (Apparently this is given as an option in case someone has manually changed an automatically generated ToC, which is almost always a really bad idea.)   Unfortunately, "Update page numbers only" is the default selection in the dialog, which means that I have to select the other option every time. I update the fields all the time when I'm working in a document that has figures and cross-references, so this is an annoying extra step I'd really like to get rid of.   Is there any way to tell Word to stop asking me this stupid unnecessary question? I want to initialize a struct element, split in declaration and initialization. This is what I have:  typedef struct MY_TYPE {   bool flag;   short int value;   double stuff; } MY_TYPE;  void function(void) {   MY_TYPE a;   ...   a = { true, 15, 0.123 } }   Is this the way to declare and initialize a local variable of MY_TYPE in accordance with C programming language standards (C89, C90, C99, C11, etc.)? Or is there anything better or at least working?  Update I ended up having a static initialization element where I set every subelement according to my needs. Unidentified address in Gmail Many times, when I copy&amp;paste an email address into the "To" field in a Gmail message, it refuses to send the email, saying that “the address has not been identified”. This is very frustrating. Is there any solution other than just manually typing the address? I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?  The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful.  I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation. Prove the Probability of Two Events I'm taking my first course in Probability and one of my homework problems is to prove that for any two sets:  $$P(A \cup B) = P(A) + P(B) - P(A \cap B) $$  Note that the function $P$ is the probability of an event happening.  From the third axiom, we are given that for any two disjoint sets, that is $ A \cap B = \emptyset$:  $$P(A \cup B) = P(A) + P(B)$$    My thought process follows from this third axiom, but I run into a problem near the end:  Since $A$ and $B$ are disjoint sets, $P(A)=P(A-B)$ and $P(B)=P(B-A)$, so  $$P(A \cup B) = P(A) + P(B)=P(A-B)+P(B-A)$$  From this, it appears that we are essentially taking the symmetric difference of A and B.  This, to me, seems to be equivalent to:  $$P(A-B)+P(B-A)=P(A)+P(B)-P(A \cap B)$$  Hence, by my logic:  $$P(A \cup B) = P(A)+P(B)-P(A \cap B)$$  Which, in essence, doesn't care whether two sets are disjoint or not as we are subtracting the intersection of them.  If the sets are disjoint, then the intersection is merely null.  If they intersect, then we are removing the "extra pieces".  I feel like this is inadequate proof despite my ability to convince myself that it is true..  Where, if anywhere, did I go wrong?  Thanks! How does a Support Vector Machine (SVM) work?