Integrals of Powers of Secant and Tangent

Use tangent as the substitution variable when the power of secant is even. Since the derivative of tangent is secant squared, you can factor out sec²x from the integrand. The remaining even power of secant is then converted entirely into tangent terms using the identity sec²x = 1 + tan²x, transforming the integral into a polynomial that can be integrated directly.

When the power of tangent is odd, choose secant as the substitution variable. Factor out sec(x)tan(x) from the integrand, which matches the derivative of secant. Rewrite the remaining even power of tangent in terms of secant using the identity tan²x = sec²x - 1. This converts the integral into a polynomial expression that can be integrated using standard algebraic techniques.

The identity sec²x = 1 + tan²x is used to rewrite even powers of secant entirely in terms of tangent. This identity allows you to eliminate secant from the integrand after factoring out sec²x, leaving only tangent terms. The resulting polynomial in tangent can then be integrated using standard integration rules before converting back to the original variable.

Substitution works because trigonometric identities allow you to rewrite all remaining powers of one function in terms of the substitution variable. Once the integrand contains only the new variable, it becomes a polynomial expression. Polynomials are straightforward to integrate algebraically, and the result is then converted back to trigonometric functions, similar to techniques used in integrals of powers of sine and cosine.

For even powers of secant, tangent is the substitution variable because sec²x appears in its derivative. For odd powers of tangent, secant is the substitution variable because sec(x)tan(x) appears in its derivative. In both cases, the strategy pairs part of the integrand with a derivative and uses trigonometric identities to rewrite remaining factors, simplifying the integral into polynomial form.

Direct integration of secant-tangent powers is difficult because the functions are transcendental. Substitution paired with trigonometric identities transforms the integral into a polynomial, which is much easier to integrate. After integration, the polynomial result is converted back to trigonometric functions, providing a systematic approach that works for any combination of even secant or odd tangent powers.

Once the integral is converted to polynomial form in the substitution variable, apply standard integration rules to integrate each term. After obtaining the antiderivative in terms of the new variable, substitute back to express the result in terms of the original trigonometric function. This final conversion completes the solution and returns the answer to the original variable form.