Ch. 7: Integration

Question 1

How does u-substitution work?

Answer 1

When you have an integral that looks like the output of the chain rule,

Substitute the inner function for just a variable to make the whole process easier

Question 2

When should you use the method of partial fractions?

Answer 2

When we have a rational expression where we can factor the denominator.

Question 3

(Partial Fractions)

Integral of (x+5) / (x^2 + x - 2) dx.

How to solve?

Answer 3

Break denominator into factors
Find numerators by letting X = various things
Integrate separately using u-sub

Question 4

What are the three ‘structures’ under which you can solve partial fractions?

Answer 4

Linear factors

Irreducible quadratic

Repeated linear factors

Question 5

What is the derivative of arctan(x)?

Answer 5

1/(x^2+1)

Question 6

What is the derivative of sec(x)?

aka 1/cos(x)

Answer 6

tan(x)*sec(x)

= tan(x) * (1/cosx)
=(sinx/cosx) * (1/cosx)
=sinx / cos^2(x)

Question 7

What is sec^2(x) - 1?

Answer 7

tan^2(x)