Integrals

Integrals

1. Can I directly integrate?

   • Lol of course not
   • Try your trig identities, your 1/x = ln x, and other standard forms

2. Can I use substitution?

   • Try to replace part of f with u, and another part with du
   • Ha. Nope.

3. Can I do integration by parts?

   • uv - {v du (be prepared to do it twice!)
   • No can do.

4. Does it look like a rational integral? (P/Q)

   • If degree P >= degree Q: long division!
   • If degree P < degree Q: factor the denominator and split into two (solve for A and B)
   • None of these work!
     • Do we have a partial fraction?
       • Again with the splitting and finding A/B!
     • How about a repeated root?
       • Split up again, this time with A and Bx + C

5. Do we have an improper integral?

   • If there are infinities, replace with lim R -> ...
   • If it's more like some c, replace with lim c -> ...

6. Are we asking existential questions?

   • Compare and find upper bound

7. Does it look like trig substitution?

   • Do we have sqr(a^2 - x^2)? Replace x with asin0
   • See a^2 + x^2 anywhere? x = atan0
   • Must be sqrt (x^2 - a^2)! x = asec0
   • Still no? Some other inverse substitution then.

8. Are you being asked to approximate?

   • Remember trapezoid rule, midpoint rule, and Simpson's rule? Yeah, those. (And don't forget those error estimates)

9. Is area being mentioned?

   • Between curves? You're lucky, just subtract and integrate!
   • They want surface area? Better plug in 2pi{x sqrt(1 + f'(x)^2) for y-axis, and replace the x with f(x) for x-axis

10. Volume is the goal here.

    • Figure out if you want discs (pir^2) or shells (2pirh)

11. Arc length? That's a rare one.

    • Plug in {sqrt(1 + f'(x)^2) then.