POWERS OF TRIGONOMETRIC FUNCTIONS

The integrals of powers of trigonometric functions will be limited to those which may, by substitution, be written in the form

EXAMPLE: Evaluate

SOLUTION: Let

so that

By substitution,

Then, by substitution again, find that

Therefore,

EXAMPLE: Evaluate

SOLUTION: Let

u = cos x

so that

du = -sin x dx

We know that

so by substitution

PRACTICE PROBLEMS: Evaluate the following integrals:

ANSWERS:

SUMMARY

The following are the major topics covered in this chapter:

1. Integral of a variable to a power: The integral of a variable to a power is the variable to a power increased by one and divided by the new power.

Formula.

2. Integral of a constant: A constant may be written either before or after the integral sign.

Formula.

3. Integral of the sum of differentiable functions: The integral of an algebraic sum of differentiable functions is the same as the algebraic sum of the integrals of these functions taken separately.

Formula.

4. Integral of a function raised to a power: The integral of a function raised to a power is found by the following steps:

1. Increase the power of the function by 1.

2. Divide the result of step 1 by this increased power. 3. Add the constant of integration.

Formula.

5. Integral of quotients:

Method 1. Integrate by putting the quotient into the form of the power of a function.

Method 2. Integrate quotients by use of operations of logarithms.

Formula.

Method 3. Integrate quotients by changing the integrand into a polynomial plus a fraction by dividing the denominator into the numerator.

6. Integral of a constant to a variable power:

Formula.

where u is a variable, ais any constant, and e is a defined constant.

7. Integral of trigonometric functions:

8. Integral of powers of trigonometric functions: The integrals of powers of trigonometric functions will be limited to those which may, by substitution, be written in the form J u" du.

ADDITIONAL PRACTICE PROBLEMS Evaluate the following integrals:

ANSWERS TO ADDITIONAL PRACTICE PROBLEMS