Calculus question?

Question

The math problem i have is on this site:http://calcchat.tdlc.com/free_solutions/main.html
Textbook:8th edition, Chapt:8, Sect:4, and Excercise:13

This site shows the solution to the math problem in my textbook, but in excercise #13 i dont get how they went from step 5 to step 6. This problem deals with trig substitution.
PLz go on that site and explain how they calculated those steps.

Answer 1

Ok, so you make the substitution.

So you at at step 5. int(tan[t]sec[t]sec^2[t])
Now solve this trig integral with a U substitution (Oh Jolly what fun)
5a) U = sec(t) dU = tan[t]sec[t] (Remember form last semester? If not just take derivative of 1/cos[t] or trust me)
Lookie lookie we have tan[t]sec[t]
5b)int(U^2 du) = U^3/3
5c) back sustitute U = sec(t) U^3/3 = sec^3(t)/3

That'll just about get you to step 6.

Answer 2

∫sec θ tan θ sec² θ dθ
u=sec θ, du=sec θ tan θ dθ
∫u² du (the sec θ tan θ cancels neatly)
u³/3 + C
sec³ θ/3 + C

And since sec θ = √(1+x²), we have:

(1+x²)^(3/2)/3 + C

Note that it would have been much simpler, and faster, to evaluate the integral using the substitution u=1+x², du=2x dx, thus:

∫x√(1+x²) dx
∫√u/2 du
u^(3/2)/3 + C
(1+x²)^(3/2)/3 + C

Answer 3

Whoops my bad. I thought it said Cactus question.