Trigonometric Substitution
The object of trigonometric substitution is to eliminate the radical in the integrand or the thing it is that you're integrating. This is possible because of the Pythagorean identities. Note that the square root sign is not an html character. As a result SQRT will denote the square root sign.
Pythagorean identities
(cos A)2 = 1 - (sin A)2
(sec A)2 = 1 + (tan A)2
(tan A)2 = (sec A)2 - 1
Trigonometric Substitution (a > 0)
1. For integrals involving SQRT(a2 - u2), let u = a sin A.
Then SQRT(a2 - u2) = a cos A where -pi/2 < A < pi/2.
2. For integrals involving SQRT(a2 + u2), let u = a tan A.
Then SQRT(a2 + u2) = a sec A where -pi/2 < A < pi/2.
3. For integrals involving SQRT(u2 - a2), let u = a sec A.
Then SQRT(u2 - a2) = ± a tan A where 0 < A < pi/2 or pi/2 < A < pi. Use the positive value if u > a and the negative value if u < -a.
