Question

Evaluate the integrals in Exercises 33–52.∫ sec³(x) tan³(x) dx

Accepted Answer

Start by rewriting the integral $$ \int \sec^{3}(x) 	an^{3}(x) \, dx  to $$express powers of tangent and secant in a form that allows substitution or use of identities. Recall that $$ 	an^{3}(x) = 	an^{2}(x) 	an(x) . $$Use the Pythagorean identity $$ 	an^{2}(x) = \sec^{2}(x) - 1  to $$rewrite $$ 	an^{3}(x)  as$$ $$ (\sec^{2}(x) - 1) 	an(x) . $$Substitute this back into the integral to get $$ \int \sec^{3}(x) (\sec^{2}(x) - 1) 	an(x) \, dx . $$Recognize that $$ 	an(x) \, dx $$ can be related to the derivative of $$ \sec(x) $$, since $$ \frac{d}{dx} \sec(x) = \sec(x) 	an(x) . $$This suggests a substitution: let $$ u = \sec(x) $$, so that $$ du = \sec(x) 	an(x) \, dx . $$Rewrite the integral in terms of $$ u $$ and $$ du . $$Notice that $$ \sec^{3}(x) (\sec^{2}(x) - 1) 	an(x) \, dx = u^{3} (u^{2} - 1) 	an(x) \, dx . $$Using the substitution, express $$ 	an(x) \, dx  in $$terms of $$ du $$ and $$ u . $$After substitution, simplify the integral to a polynomial in $$ u $$, then integrate term-by-term with respect to $$ u . $$Finally, substitute back $$ u = \sec(x)  to $$express the answer in terms of $$ x .$$

Evaluate the integrals in Exercises 33–52.
∫ sec³(x) tan³(x) dx

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Key Concepts

Trigonometric Identities

Integration by Substitution

Reduction Formulas for Powers of Secant and Tangent