Solving Secant and Tangent Integrals

Integrals of secant and tangent powers are often solved by choosing a trig substitution that matches the integrand. The key idea is to use the parity of the exponents, which means whether a power is even or odd. A good substitution lets part of the integrand pair with the derivative of a trig function.

Integrals involving powers of tangent and secant can be solved using substitution, and the approach depends on whether the power of secant is even or the power of tangent is odd.

If the power of secant is even, choose tangent as the new variable, because its derivative is secant square, which can be factored out.

The remaining even p...