INTEGRATING PRODUCTS OF TANGENTS AND SECANTS

Consider the following integral :

Case 1 :

If q is a positive even positive integer, substitute 

u = tanθ

and evaluate the integral by substitution method.

Case 2 :

If both p and q are positive odd integers, substitute 

u = secθ

and evaluate the integral by substitution method.

You may have to use the following trigonometric identies.

sec²θ = 1 + tan²θ

tan²θ = sec²θ - 1

Let u = tanθ.

ᵈᵘ⁄dθ = sec²θ

du = sec²θdθ

Let u = tanθ.

ᵈᵘ⁄dθ = sec²θ

du = sec²θdθ

Let u = secθ.

ᵈᵘ⁄dθ = secθtanθ

du = secθtanθdθ

Let u = secθ.

ᵈᵘ⁄dθ = secθtanθ

du = secθtanθdθ

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