When integrating products of secants and tangents, we are integrating expressions that are in the form:
To integrate products of secants and tangents, the key is to utilize trigonometric identities and strategic substitutions.
If the power of secant is even, use the identity sec²x = tan²x + 1 and substitute u = tan(x). If the power of tangent is odd, use the identity tan²x = sec²x - 1 and substitute u = sec(x).
Secant and tangent are trigonometric functions that relate the angles of a right triangle to the ratios of its sides.
If the integral has a factor of sec²x, let u = tan(x).Rewrite the remaining secant terms using the identity sec²x = 1 + tan²x and then substitute u and du = sec²x dx to simplify the integral to a polynomial in u
Let u = sec(x) if the integral has an odd power of tangent. Then, rewrite the remaining tangent as tan²x = sec²x - 1. You then substitute u and du = sec(x)tan(x) dx to simplify the integral.
There are several cases to consider in this type of integration.
case 1 Integrating products of secants and tangents
This is a case where m is an odd positive integer . In this case we split off the sec(x)tan(x) to form a differential sec(x)tan(x) of sec(x) along with dx.
We then use the identity ‘sec2x = 1-tan2x‘ to convert the remaining power into powers of sec(x). This way, we prepares the integrand for the substitution of u = sec(x). consider the following:
u = sec(x)
hence our earlier equation becomes;
substituting;
case 2 of Integrating products of secants and tangents
This is a case where n is an even positive integer. We split sec2x to form a differential of tan(x) along with dx. We then use the identity ‘sec2x = 1 + tan2x’ to convert the remaining even powers of x into powers of tan(x). This prepares the integrand for substitution in u = tan(x). consider:
u = tan(x) and du = sec2xdx