Trigonometric Integrals: Guidelines for tan(x)cos(x)

[mes]

In this video I go over a strategy for solving integrals of the form tan(x)^mcos(x)ⁿ where m and n are integers and both are greater than or equal to 0. I go over 2 specific cases and are:

(1) n is even
(2) m is odd and n is greater than 0

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Trigonometric Integrals: Guidelines for tan(x)sec(x)

In my earlier video I summarized guidelines for solving integrals of the form:

Where m and n are integers and m, n ≥ 0

Similarly, the summary of the guidelines for integrals of the form

is shown below:

Strategy

(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 tan x:

Then substitute u = tan x because du = sec²x dx

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

Then substitute u = sec x because du = sec(x)tan(x)dx

(c) If the power of secant is even AND the power of tangent is odd, you can use both steps (a) or (b) but usually step a) would be easier.