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The Natural Logarithmic Function: Integration

Objective: Use the Log Rule for Integration to integrate a rational
function. Integrate trigonometric functions.

The differentiation rules

and

Log Rule for Integration
Let u be a differentiable function of x

Because du = u′dx the second formula can be written as

Ex: Evaluate:

Ex: Find

Ex: Find the area of the region bounded by the graph
, the x-axis, and the line x = 3.

Ex:

Ex: (use long division first)

Ex:

Guidelines for Integration

1. Learn a basic list of integration formulas.

2. Find an integration formula that resembles all or part of
the integrand, and, by trial and error, find a choice for
u that will make the integrand conform to the formula.

3. If you can’t find a u-substitution that works, try altering
the integrand. Try a trigonometric identity,
multiplication or division. Be creative.

4. If you have access to computer software that will find
the antiderivative symbolically, use it.

Ex: Find

Integrals of the Six Basic Trig Functions

Ex: Evaluate: