Week 7: Methods of Integration

Question 1

When there is a change in variables for a definite integral what must also be changed?

Answer 1

When the variable of a definite integral is changed, the limits of the integral must also be changed.

Question 2

What is the formula used to apply integration by parts?

Answer 2

∫u.dv = uv - ∫v. du

Question 3

How do you remember how to choose which function is better for “u”?

Answer 3

LIATE. 
Logarithmic
Inverse trigonometry
Algebraic polynomial
Trigonometric
Exponential

Question 4

When using integration by parts, if an integral is continuous (i.e. that is the you must use integration by parts multiple times and the integral will never disappear), how do you determine I?

Answer 4

You must continue to apply integration by parts multiple times until you get the original integral which is equal to I. When you get the original integral, replace with I and then arrange the equation for I as the subject.

This will happen, generally when working with trigonometric or exponential functions.

Question 5

How do you deal with integrals with quadratic denominators that have no linear factors?

Answer 5

You must manipulate the numerator into something that involves the derivative of the denominator.
i.e.
k[ f’(x) ] + something
This manipulation allows us to us the fact that a log functions derivative is “derivative over original”.