Now that we have learned strategies for solving integrals with factors of sine and cosine we can use similar techniques to solve integrals with factors of tangent and secant. Using the identity sec²x = 1 + tan²x we are able to convert even powers of secant to tangent and vice versa. Now we will consider two examples to illustrate two common strategies used to solve integrals of the form

Suppose we have an integral such as

Observing that (d/dx)tanx=sec²x we can separate a factor of sec²x and still be left with an even power of secant. Using the identity sec²x = 1 + tan²x we can convert the remaining sec²x to an expression involving tangent. Thus we have:

Then substitute u=tanx to obtain:

Note: Suppose we tried to use the substitution u=secx, then du=secxtanxdx. When we separate out a factor of secxtanx we are left with an odd power of tangent which is not easily converted to secant.

Consider the integral

Since (d/dx)secx=secxtanx we can separate a factor of secxtanx and still be left with an even power of tangent which we can easily convert to an expression involving secant using the identity sec²x = 1 + tan²x. Thus we have:

Then substitute u=secx to obtain:

Note: Suppose we tried to use the substitution u=tanx, then du=sec²xdx. When we separate out a factor of sec²x we are left with an odd power of secant which is not easily converted to tangent.

Strategy for Evaluating

(a): If the power of secant is even (n=2k, k>2) save a factor of sec²x and use the identity sec²x = 1 + tan²x to express the remaining factors in terms of tanx.

then substitute u=tanx.

(b): If the power of tangent is odd (m=2k+1), save a factor of secxtanx and use the identity sec²x = 1 + tan²x to express the remaining factors in terms of secx.

then substitute u=secx.

Note: If the power of secant is even and the power of tangent is odd then either method will suffice, although there may be less work involved to use method (a) if the power of secant is smaller, and method (b) if the power of tangent is smaller.

Find the indefinite trigonometric integral