Education An integral that does not like me. . .

So today in Calc the problem was taking the integral of: sec^3 x tan x

Some work (oooh, pretty. . . I love open office) if ya want to skip the image and get straight to the answer - I get (2/5)sec^5 x


So, how do I check my answer? Take the derivative - and here is where the problem reveals it self.


I think all the work has been done properly without any errors, my Calc teacher even looked at the work (though somewhat quickly) and he could not spot any obvious errors. By the way, my Calc teacher did have a different method to solve this problem - a u substitution with u = sec, and du = sec tan - which works fine, and passes the test of differentiating the answer to get the original equation.

According to my calc teacher, it is "normal" in Calc to have a problem that can be worked more than one way, and that sometimes doing all the work properly will not always yield the proper answer, as illustrated with what I did. If that is really true about Calc I have a bad feeling there will be a lot of ahead. . . Don't get me wrong, so far, I am loving Calc. Yea, I've kinda lost my mind.

Back on track - can anyone spot some sort of error I might have made? Really off topic - but hey, its my thread so I can do what I want, right? In the 1st sentence of this paragraph, at the part "error I might have made" should the "have" be replaced with "of" or what? That little bit of grammar is something that has been haunting my dreams lately. . .

L J

 

If you want to check your answer, then try using http://integrals.wolfram.com/index.jsp - it's run by the people who make Mathematica.

 

Bah, I've been doing complex and damped SHM questions since 4pm. Brains gone AWOL. I'll have a go at that tomorrow if I get some free time.

 

Oh the delights of SHM - very useful for Quantum Mech though.

 

The derivative of sec²(x) is not tan(x), it's the other way round. You should have du = 2sec²(x)tan(x) dx.

(unless I'm reading it wrong, if so then I'll have another look tomorrow. Bedtime!)

Edit:

It's correct. Saying "should of" is wrong; it comes from people mishearing "should've".

 

The derivative of tan(x) is sec^2(x). I can't help you any more, as I just got into integrals and don't know much about them yet.

 

Man this is crazy stuff, I was excellent at Maths, but was always crap with algebra etc, never got the hang of it, but I think that's because I was never taught it at a reasonable level.

 

The first bit looks fine secx^3 = secx^2*secx then saying u = secx^2 etc. Once you've done that integrate by parts using INT f.g = f*INTg - INT(df/dx*INTG)

 

Int of Sec^3(x) Tan(x) dx

First pull out a secant:
Int of Sec^2(x) Sec(x) Tan(x) dx

Then we can set Sec(x)=u:
Int of u^2 Sec(x) Tan(x) dx
u=Sec(x)
du=Sec(x) Tan(x) dx

Substitute in dx to get:
Int of u^2 du

Which is:
(1/3)u^3 + c

Substitute u back in:
(1/3)Sec^3(x) + c