All Answers 358510 Answers

It depends on the limits of the integration.  If the limits are ±infinity then the integral can be done analytically - see below.  However, if at least one of the limits is finite then the integral can only be done numerically (an exception is if one of the limits is zero, then the integral from zero to infinity is just half of the integral from -infinity to infinity).

Here is a brief derivation of the integral of e^{-x^2} from - infinity to infinity: 

$$Let $$I=\int_{-\infty}^{\infty}e^{-x^2}dx$$

Because x is a dummy variable we can also write this as $$I=\int_{-\infty}^{\infty}e^{-y^2}dy$$

Multiply the two together to get $$I^2=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2}dxdy$$

Now if we think of x and y as being axes on a standard Cartesian graph, then the right-hand side is just the integral over the whole plane of $e^{-(x^2+y^2)}$. Change to polar coordinates, $r,\theta$ where $$x = r.cos\theta, y = r.sin\theta$$.\\
The limits of the integration become $0$ to $\infty$ for $r$ and $0$ to $2\pi$ for $\theta$. $dxdy$ becomes $r.drd\theta$. So:\\$$

$$I^2=\int_{0}^{2\pi}\int_{0}^{\infty}e^{-r^2}r.drd\theta$$\\
$$I^2=2\pi\int_{0}^{\infty}e^{-r^2}r.dr$$\\
$$I^2=2\pi\int_{0}^{\infty}e^{-r^2}\frac{1}{2}dr^2$$\\
$$I^2=\pi\int_{0}^{\infty}e^{-z}dz$$\\

where I've substituted $z$ for $r^2$ in that last integral. So:
$$I^2=\pi$$
as $\int_{0}^{\infty}e^{-z}dz=1$
Finally, $I=\sqrt\pi$ or
$$\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt\pi$$$$

Some may have misread the word above. No vulgarity is intended.

Neither HOMONYM(ic) nor HOMOPHON(ic) words were intended as a vulgar pun toward CPhill – He’s too polite.

Now, if it was some asshole on here, then, yes, it would be intended.

Sisyphus was a king of Ephyra (now known as Corinth) punished for chronic deceitfulness by being compelled to roll an immense boulder up a hill, only to watch it roll back down, and to repeat this action forever.

See this: wikipedia.org/wiki/Sisyphus

Assuming this is f(x) = x - 2ln(x+1), with 0≤x≤2, the best thing to do first is to plot a graph to see what it looks like.  In this case you will see a curve with a single minimum.

Differentiate the function with respect to x and set the result to zero. 

$$1-\frac{2}{x+1}=0$$

This will give x = 1.  So the minimum occurs at x = 1 and the function has a value of f(1) = 1 - 2ln(2) there.

$${\mathtt{1}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{2}}\right)} = -{\mathtt{0.386\: \!294\: \!361\: \!119\: \!890\: \!6}}$$

Hi Stu,

It is nice to see you again.

Is there supposed to be a - sign in front of the f(x) or is that just a dash?

In answer to all those people who want to know how to do fractions on this calculator:

Here are some of the replies from various answers; I hope they can help you out:

Re: How do you do fractions on this calculator?

"5 and a half - 8 and a quarter would be

$$\left({\mathtt{5}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,-\,}}\left({\mathtt{8}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{4}}}}\right) = -{\mathtt{2.75}}$$ "(by Melody)

Re: where is the fraction button

"It is a box above a box" (by Anonymous)

"On many calculators out guest is correct. There is no fraction button on the Web2.0calc  You have to use the $$\div$$ button. To enter 3 and 1/2 - 6 and a quarter you would press

$$(3+1\div2)-(6+1\div 4)=$$" (by Melody)

Re: How do we use fractions on this calculator??

"with the $$\div$$ button. If you are doing calculations with mixed numerals it is best to put them in brackets like $$(3+1\div 2)$$" (by Melody)

"You can use forward slash / as we write in a fraction. 6/2 is a fraction. And if there are more numerals involved, use brackets as (6*2+8)/4

I hope it helps." (by novice)

"you simply make it a decmil if 4 were 1/4 i would put 4.25" (by Gman2598)

Thanks to everyone who answered!

$$\frac{d}{dx}x=1$$

In other words the gradient of the line y=x is 1

I don't get it.  

Thanks Rom,

I'd like to get back and have a proper look at this one!   

hi Professor,  It is great that you answered this question.  I love people to get involved.

Please edit your last step to make it correct, I am reluctant to give you points until it is fixed.   

Let me add something to what kitty<3 said.

Ask youself the question.....what could "y" be ??

Notice that, if we square any positive or negative "x," we get a "positive."

And, if we square "0," we get "0."

So, as kitty<3 points out, "y" has to be equal to or greater than 0.

First of all I'm assuming you mean the range of f(x) = x²  

Think of it this way: the range is the set of all the y-values that fit on this graphed function, which is a parabola. You can graph it to make things clearer:)

This parabola is the parent function (the basic parabola function.)

It is a positive function (opens upward) and has its vertex at (0,0).

Since the vertex (lowest point in this case) is at y=0, the y-values cannot go lower than 0.

Therefore, while the domain would be all real numbers, the range would be numbers greater than/ equal to 0.

Great answer, CPhill!

I just want to point out something to Anonymous:

Your reasoning is good:) but you forgot one small detail. 2√2 is equal to √8, not 8. The square root sign makes a big difference!

$${\frac{{\mathtt{4}}}{{\mathtt{7}}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{49}}}{{\mathtt{36}}}}\right){\mathtt{\,-\,}}{\frac{{\mathtt{8}}}{{\mathtt{9}}}} = -{\mathtt{0.111\: \!111\: \!111\: \!111\: \!111\: \!1}}$$

What do you mean.

If you want inverse tan it is just 2nd tan

It may help if I tell you that inverse tan is the same as arc tan

tan^-1 = atan

This page should help you 

That is what Alan said too but I always use a little supertext c to represent radians.

Here are some arrow comands that may be useful

I am sorry Daajkoma but I don't know that anyone on this forum has the knowledge to answer this question.

392. You should get a calculator, or use the one on the computer.

Mr.Henry is painting a square floor of side 36m. He divided the floor into 12 equal sections.

A.He painted 4 sections. What percentage of the floor is painted?

B.Calculate the area of the floor?

C. What area of the floor is not painted.Give answer in sp metres.

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A. 33% .......(4 sections represent 1/3 of the total = 33%)

B. Area of floor = 36m X 36m = 1296 sq meters

C. If 1/3 is painted, 2/3 must not be. So, 2/3 x 1296 = 864 sq meters !!

I'm not sure whether you meant 80% of 788, or if you meant 80% off of 788. Either way, this should help.

Multiply 0.80 by 788. That will tell you how much is off the original; 80% of 788. To find the new amount, subtract the amount off from 788.

0.80 x 788 = 630.4 (this is 80% of 788)

788 - 630.4 = 157.6 (this is the new cost once you have subtracted how much is off of it)

s √2+√2 = 2√2 or √4

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This kind of stuff use to confuse me, too.  Here's a hint....when confronted by something that you're not sure of .... use some numbers that you know and see if this leads you to the correct answer !!

For instance ......is √9+√9 = 2√9 or √18   ???

Well....√9+√9 = 3 + 3 = 6

And....... 2√9 = 2*3 = 6

Mmmmm...that looks a lot like the first answer, doesn't it??

Think about √18.....the √16 = 4 and the √25 = 5...so the √18 must be between 4 and 5 and that's certainly < 6 !!

Can you answer your own question, now??

Its neither because 2√2 is 8 and √4 is 2. √2+√2 is 2.8284271247461901, as for what it can be equal to, i dont know.

Alan's answer is correct.....Here's a slightly different approach

I first put "g" into "f"

So we have.....

-2(-4x + 6) - 7    =

8x - 12  - 7  =

8x - 19

Now, put the (-5) into this "composite" function

We have:

8(-5) - 19   =

-40 - 19  =

-59

Either approach works...your choice!!