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**preview**shows half of the first page. to view the full**1 pages of the document.**7.4 Integration Techniques

Partial Fractions for Rational Functions

Question #5 (Medium): Rationalizing Substitutions

Strategy

Non-rational functions refer to those containing root functions in either the numerator or denominator,

but also those that include trigonometric or exponential functions. Either way, a simple variable is used

to substitute as the building block to represent the function type. Then factor the denominator, split the

rational function, then take integral. Remember to re-substitute back in the original expression.

Sample Question

By substitution express the integral as a rational function and then evaluate the integral.

Solution

First factor the function in the denominator, so that: .

Notice the rational function is in the form of chain rule. So let and . Then:

. Apply partial fractions:

. When , then , so

.

means

, so

.

Then:

Substitute back in the original expression:

Therefore, the integral is evaluated as:

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