Jan 26

Expression of the form a2-b2⁢x2 a squared minus b squared x squared : We make a sine substitution

Substitute x=ab⁢sin⁡θ x equals a over b sine theta (-π2≤θ≤π2 open paren negative pi over 2 is less than or equal to theta is less than or equal to pi over 2 so dx=ab⁢cos⁡(θ)⁢dθ d x equals a over b cosine theta d theta and then

a2-b2⁢x2 the square root of a squared minus b squared x squared

=a2-b2⁢(ab⁢sin⁡θ)2 equals the square root of a squared minus b squared times open paren a over b sine theta close paren squared

=a2-a2⁢sin2⁡θ equals the square root of a squared minus a squared sine squared theta

=a⁢1-sin2⁡θ equals a the square root of 1 minus sine squared theta

=a⁢cos2⁡θ equals a the square root of cosine squared theta

=a⁢cos⁡θ equals a cosine theta