Integral of Secant Squared over Tangent Substitution

[Schrodinger's Dog]

Homework Statement

$$\int_\frac{-\pi}{4}^\frac{\pi}{4} \frac{sec^2(x)}{(\sqrt{1-tan^2(x)})}$$

Homework Equations

None

The Attempt at a Solution

Ok I've done the substitution u=tan(x) and it neatly works out as

$$\int_\frac{-\pi}{4}^\frac{\pi}{4} \frac{1}{(\sqrt{1-u^2})}\rightarrow sin^{-1}(tan(x))+c$$

I get 3.1416 to 4dp? Can someone check that is right, I've checked it on my calc, but when I type it into the maths program I have it throws up an error message for some reason. Just want someone to confirm I have the correct figure and that I haven't done something silly.

 

[bob1182006]

Yes the answer's pi.
maybe you got the error message because when doing the substitution and changing the limits the fraction is undefined for 1?

but if you go directly from the question to the answer w/o substituting you'll get pi as the answer.

 

[Schrodinger's Dog]

Thanks for the advice and the check

It won't evaluate any value of it.

It gives the answer to the general integral as:-

$$\frac{1}{tan^4} \frac{\left [-2(1-tan^2(x))^{\frac{1}{2}}+\frac{2}{3}(1-tan^2(x)^\frac{3}{2}\right ]}{cos}$$

Er yeah thanks nicely simplified. And when I try evaluating it with x:=pi/4 and x:=-pi/4 or any value for x for that matter it won't touch it. Apparently it finds it's own answers to be unsolvable because of an error

 

[Gib Z]

If you wish to waste your life, differentiate that expression :) If you don't want to be an idiot, then realize that calculators are fallible and not as smart as we like to think they are.

It saddens me that people have more faith in their calculator than their own logic and the work of mathematicians for over 2000 years. I like to show this to people by first showing them a hand proof of the irrationality of 2, then putting in a sufficiently accurate fraction, squaring it and reading the calculators response of 2.

Pitifully, they go looking for an error in my proof...

 

[cristo]

If you wish to waste your life, differentiate that expression :) If you don't want to be an idiot, then realize that calculators are fallible and not as smart as we like to think they are.

:rofl: nice quote!

 

[Schrodinger's Dog]

If you wish to waste your life, differentiate that expression :) If you don't want to be an idiot, then realize that calculators are fallible and not as smart as we like to think they are.

It saddens me that people have more faith in their calculator than their own logic and the work of mathematicians for over 2000 years. I like to show this to people by first showing them a hand proof of the irrationality of 2, then putting in a sufficiently accurate fraction, squaring it and reading the calculators response of 2.

Pitifully, they go looking for an error in my proof...

Well I certainly didn't have faith in the answer alone, my calculator has done some pretty weird things in the past and that mess there is not technically a calculators answer but a maths program on my pc, which naturally I thought was full of crap. If I had that much faith I would of accepted pi as the answer, it seems reasonable, but I wanted to be sure that not only was my working correct, but the calculator or more likely I hadn't typed in something weird and really the answer was pi/2.

as to why Mathcad will not except any value of x in the solution to it's own integral is anyone's guess. It's useful but sometimes it's idea of simplicity and answers and errors could only have come from a computer algorithm.

By the way does anyone know of any resource on line that does definite integrals, I know several that do indefinite ones? Just in case my software decides to go funny again.

 

[neutrino]

By the way does anyone know of any resource on line that does definite integrals, I know several that do indefinite ones? Just in case my software decides to go funny again.

(%i2) integrate((sec(x))^2/sqrt(1-(tan(x))^2),x,-%pi/4,%pi/4);
(%o2)                                 %pi

That was from maxima, if you're interested. It's an off-line app., but pretty good.

 

[Päällikkö]

That was from maxima, if you're interested. It's an off-line app., but pretty good.

eg. http://wiki.axiom-developer.org/SandBoxMaxima offers an online version of maxima.