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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

Ex: Find

 Integrals of the Six Basic Trig Functions

Ex: Evaluate: