Calc 2 Review: Integration By Parts

LEVEL 1: Basic Integration by parts

This is the most basic integration by parts problem that there is. In this case, the hardest part is not messing up the sign after during the trig integration.

LEVEL 2: Integration by parts, the sequel

Doing integration by parts once does not get us to answer. We have to do integration by parts on the second term of the right-hand-side.

 Substituting back into the first time we did integration by parts, we get:

We finally get an answer.

We could have sped up the process by using the Tabular Method.

LEVEL 3: The neverending integration by parts.

Step One: Swear. (This step is Optional)
Step Two: Remember back to Algebra I.
   Given the equation:   a = b – 4a     Solve for a
   That’s easy. Add 4a to both sides, get 5a = b
   Divide both sides by 5, get a = b/5
   Piece of Cake.

Step Three: After reminiscing, return to the problem at hand and try integration by parts, twice.

Step Four: Remember that side trip we took back at Step Two. Our equation right now looks something like:

We are trying to solve for the integral. It's the same form as a = b - 4a

In this case, a is the integral, b has two ugly terms and instead of having 4 as a coefficient on a we have a coefficient of 49/169. We solve it the same way that we did the Algebra I problem.

Don't forget to add the + c. It would be a pity to do all the hard work and lose a point because of the + c.

SUMMARY

When using integration by parts on the repeating functions (Like the "Level 3" problem), you should only have to do integration by parts twice. If you find that you have done integration by parts three or four times, you are likely headed in the wrong direction.

LIATE is the rule of thumb for doing integration by parts. When trying to pick which term to differentiate (what the u is). Usually Pick in this order:
L - Logarithmic functions: ln(x)
I - Inverse trig functions: arcsin(x)
A - Algebraic functions: (i.e. polynomials)
T - Trig functions: sin(x)
E - Exponential functions: e^x

You should be able to do any similar problems (Level 1, 2 or 3) by the time the first exam comes around. Integration by parts will return in Section 3.6 and Section 4.4, so you want to master this skill.