Integrals of Powers of Secant and Tangent

Integrals involving powers of tangent and secant are commonly evaluated using substitution, with the strategy determined by the parity of the exponents. The method relies on pairing part of the integrand with the derivative of a suitable trigonometric function and rewriting the remaining factors using trigonometric identities.

When the power of secant is even, tangent is chosen as the substitution variable. Since the derivative of tangent is secant squared, a factor of sec⁡²x can be separated from the integrand. The remaining even power of secant is then rewritten entirely in terms of tangent using the identity sec⁡²x = 1+tan⁡²x. This converts the integral into a polynomial in the new variable, which can be integrated using standard algebraic techniques. After integration, the result is expressed back in terms of the original variable.

When the power of tangent is odd, secant is used as the substitution variable. In this case, a factor of sec⁡(x)tan(⁡x) is separated from the integrand, matching the derivative of secant. The remaining even power of tangent is rewritten in terms of secant using the identity tan⁡²x = sec⁡²x−1. This substitution again transforms the integral into a polynomial expression that can be integrated directly.

In both cases, the use of appropriate substitutions and identities simplifies the original trigonometric integral into a manageable polynomial form, allowing for straightforward evaluation and conversion back to trigonometric functions.