Thursday, 30 June 2016

calculus - The integral of sec4(x)tan(x)



Consider the integral

sec4(x)tan(x)



Now right off the bat I see two ways of solving this.




  1. Let u=sec(x)



2.Use integration by parts




Now doing the first way results in the integrand looking like
u3du=14sec4(x)+C



Which is correct but it's not the answer I'm looking for, so instead we'll do it the second way.



sec2(x)sec2(x)tan(x)dx
(tan2(x)+1)sec2(x)tan(x)dx
sec2(x)tan3(x)+sec2(x)tan(x)dx
Now this is where I got stuck, because I don't know whether to continue with Pythagorean identities or to factor a term out and solve for that. Or perhaps even break the two up and create two integrals.


Answer




Write your integrand in the formtan(x)(tan2(x)+1)sec2(x) and substitute u=tan(x) and you will get u(u2+1)du


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...