1. Ch. 7 – Differential Equations and Mathematical Modeling
7.2 – Antidifferentiation by Substitution

2. Although we can differentiate almost any function manually, one cannot antidifferentiate every function manually!
When integrating a function inside of another function, you must consider using substitution!
Ex: Evaluate .
We should know the integral of cosx , but not cos2x, by memory
Let’s substitute u in for 2x to make the problem easier to integrate…
…however, we want du, not dx, so use a derivative to substitute for dx as well! Since u = 2x…
Lastly, substitute the variable expression back in for u!

3. Ex: Evaluate . Try 1: Let u = x3 Try 2: Let u = x3 - 2
The tricky part is knowing what to substitute for. We’ll try different substitutions until we get the right one…
Try 1: Let u = x3
We could antidifferentiate this function, but it isn’t one that you’ve memorized how to do because of the u-2 part inside the square root. Try again, with a small change…
Try 2: Let u = x3 - 2

4. Ex: Evaluate . Try 1: Let u = sinx Try 2: Let u = ecosx
This is terrible! We don’t want x’s and u’s in the same equation! Try again…
Try 2: Let u = ecosx
Rats! We still have x’s and u’s in the same equation! Try again…
Try 3: Let u = cosx

5. Ex: Evaluate . I’ll give you a hint: write cotx as cosx/sinx…
Try 1: Let u = cosx
A u and an x in the same integral! Hot garbage! Try again…
Try 2: Let u = sinx

6. Ex: Evaluate . Try 1: Let u = xlnx Try 2: Let u = lnx
Use Product Rule!
A u and an x in the same integral! Horsefeathers! Try again…
Try 2: Let u = lnx
No absolute value bars necessary because lnx is always positive!

7. Ex: Evaluate . Try 1: Let u = x Try 2: Let u = x2
This has gotten us nowhere, for we have the same problem that we started with.
Try 2: Let u = x2
Cheese and crackers! We can’t have x’s and u’s in the same integral!
There aren’t any more choices for u, so this problem cannot be integrated by substitution! This may happen sometimes!

8. Ex: Evaluate . Try 1: Let u = 1 + x3/2
Substitute back in for u before evaluating your bounds!