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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

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