6. Integration: Inverse Trigonometric Forms

by M. Bourne

Using our knowledge of the derivatives of inverse trigonometric identities that we learned earlier and by reversing those differentiation processes, we can obtain the following integrals:

math expression

Example 1: Integrate: math expression.


Answer

Example 2: Integrate: math expression


Answer

Example 3: Find the area bounded by the curve math expression and the lines x = 0, y = 0 and x = 2.


Here is the LiveMath solution to this problem:

LIVEMath


Answer

Caution:

There are a number of integrals of forms which look very similar to the above but are actually different, e.g.

math expression

We will develop methods to solve these in a later section. (See Integration by Trigonometric Substitution.)

 

Exercises

Integrate each of the given functions:

1. math expression


Answer

2. math expression


Answer

3. math expression


Answer

4. Find the area bounded by the curve math expression and the lines x = 0, y = 0 and x = 1.


First, the LiveMath solution to this problem:

LIVEMath


Answer

This is not the answer ... it is a corruption of a .gif file containing the answer!!:

math expression

[This happened in MS Word during a period of great instability - the computer's and mine...]



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