Integration

The Techniques of Integration

Table of Contents

The Fundamental Theorem of Calculus………………………………………………………..3

The Net Change Theorem…………………………………………………………………………….6

Integration by “U” Substitution……………………………………………………………………7

Integration by Parts………………………………………………………………………………….10

Trigonometric Integration………………………………………………………………………..13

Integration by Trigonometric Substitution………………………………………………..16

Integration of Rational Functions by Partial Fractions……………………………….19

Table of Integration Formulas (General Forms)…………………………………………22

Summary of Integration Techniques………………………………………………………….23

Section 5.3 – The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus gives the precise inverse relationship between the derivative and the integral. There are two parts to the Fundamental Theorem of Calculus.

The first part (FTC 1) deals with the functions in the form of

gx=axftdt

Where f is a continuous function on [a,b] and x is between a and b. So, FTC1 tells us:

If f is continuous on [a,b], then the function g defined by

gx=axftdt , a≤x≤b

Is continuous on [a,b] and differentiable on a,b and g'x=f(x).

We use FTC1 in questions to solve for the derivative ofg(x).

Example 1: Solve the derivative of gy=2yt2sintdt.

Solution: According to FTC1, g'y=f(y), and therefore the answer is simply y2siny.

Example 2: Solve the derivative of hx=21xtan-1tdt.

Solution: This problem involves of a substitution.

Let u=1x , and therefore dudx=1x2 and dhdx=dhdududx .

So h'x=ddx21xtan-1tdt=ddu2u(tan-1tdt)dudx=(tan-1u)dudx[by FTC1]=-tan-11xx2

Tips and Tricks:

1. Notice the two questions above. We don’t need to substitute the bounds if the chain rule derivative is equal to 1 or a constant, such as in the first case (eg. If the bound is x, 2x, 3x…). In the second case, we had to substitute 1x as u. This is because you need to perform chain rule and make dudx part of the substituted equation.

2. If you find a question in the form...