Math 155 – Calculus 2

Andy Rosen

www.rosenmath.com

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Substitution

Right now, we can only integrate a small number of functions…in this chapter we will learn two methods that will expand that list.

Ex. If f (x) = (x10 – 4)4, find f (x).

Ex. 42 5x x dx

Substitution is a result of the chain rule

• u will be the stuff on the inside

• its derivative will always show up on the outside

All x’s must change to u’s before we can integrate

Ex.

Ex.

cos sinx x dx

4ln x

dxx

Ex.

Ex.

5 2tan secx x dx

23 2x dx

Ex. 1x x dx

Pract.

1.

2.

3.

3

1

1 2dx

x

2 3 1x x dx

sin xdx

x

214 1 2x c

3 2329 1x c

2cos x c

In a definite integral, you should find the antiderivative using substitution, change back to x, and then plug in endpoints.

Ex.3

1

1

5dxx

We could have changed the endpoints to u…

Integration by PartsTo find our formula, we use the product rule (don’t write

this down):

- We pick u and dv- After using formula, you still have x’s

Ex.

Ex.

cosx x dx

2secx x dx

Picking u and dv:

• You must be able to find the antiderivative of dv.

• Often, u will be a power of x.

Ex. 2 xx e dx

xxe dx

2 sinx x dx

lnx x dxPract.

1. 2.

3. 2 cos 2 sin 2cosx x x x x c

2 21 12 4lnx x x c x xxe e c