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

Evaluate each of the following integrals.

Example 1 :

Solution :

Let u = tanθ.

ᵈᵘ⁄dθ = sec2θ

du = sec2θdθ

Example 2 :

Solution :

Let u = tanθ.

ᵈᵘ⁄dθ = sec2θ

du = sec2θdθ

Example 3 :

Solution :

Let u = secθ.

ᵈᵘ⁄dθ = secθtanθ

du = secθtanθ

Example 4 :

Solution :

Let u = secθ.

ᵈᵘ⁄dθ = secθtanθ

du = secθtanθ

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