Evaluate the subsequent integral.

∫ (tan x/sec⁴ x / sec⁴ x)  dx

Solution

This kind of integral approximately falls into the form given in 3c.  It is a quotient of tangent and secant and we are familiar with that sometimes we can use similar methods or techniques for products of tangents and secants on quotients.

The procedure from that section tells us that if we have even powers of secant to strip two of them off and transform the rest to tangents. That won't able to work here. We can split two secants off, but they would be in the denominator and they would not do us any good there.  Keep in mind that the point of splitting them off is thus they would be there for the substitution u = tan x .  That needs them to be in the numerator.  Thus, that won't work and so we will have to find out another solution method.

Actually there are two solution methods to this integral depending upon how you want to go about it. We'll take a look at both.

Solution 1

In this solution technique we could just convert all to sines and cosines and see if that provides us an integral we can deal with.

∫(tan x / sec⁴ x) (dx)

= ∫ (sin x / cos x) cos⁴ x dx

= ∫ sin x cos³ x dx                                u=cos x

= -∫ u³ du

= - ¼ cos⁴ x + c

Note that just transforming to sines and cosines won't all time work and if it does it won't always work this adequately.  Frequently there will be so many works that would require to be done to complete the integral.

Solution 2

This solution technique goes back to dealing with secants and tangents.  Let us notice that if we had a secant in the numerator we could just employ u = sec x as a substitution and it would be a quite quick and simple substitution to use. We do not have a secant in the numerator.  Though, we could very easily get a secant in the numerator merely by multiplying the numerator and denominator by secant.

∫ (tan x / sec⁴ x) dx

= ∫ (tan x sec x / sec⁵ x) dx                                          u = sec x

= ∫ 1/u⁵ (du)

= - (1/4) (1/sec⁴ x) + c

 = - ¼ cos⁴ x+c