Question

Evaluate the integrals in Exercises 33–52.∫ tan⁴(x) sec³(x) dx

Accepted Answer

Start by expressing the integral in terms of powers of $$ 	an(x) $$ and $$ \sec(x) $$: $$ \int 	an^{4}(x) \sec^{3}(x) \, dx . $$Recognize that $$ \sec^{3}(x) = \sec(x) \cdot \sec^{2}(x) $$, and recall that the derivative of $$ 	an(x)  is$$ $$ \sec^{2}(x) . $$This suggests a substitution involving $$ 	an(x) . $$Rewrite the integral as $$ \int 	an^{4}(x) \sec(x) \sec^{2}(x) \, dx . $$Consider using the substitution $$ u = 	an(x) $$, which implies $$ du = \sec^{2}(x) \, dx . $$This will help transform the integral into a polynomial in terms of $$ u . $$Substitute $$ u = 	an(x) $$ and $$ du = \sec^{2}(x) \, dx $$ into the integral, yielding $$ \int u^{4} \sec(x) \, du . $$Now, express $$ \sec(x)  in $$terms of $$ u $$ using the identity $$ \sec^{2}(x) = 1 + 	an^{2}(x) $$, so $$ \sec(x) = \sqrt{1 + u^{2}} . $$Rewrite the integral as $$ \int u^{4} \sqrt{1 + u^{2}} \, du . $$This integral involves a polynomial times a square root, which can be approached by expanding or using a substitution such as $$ w = 1 + u^{2}  to $$simplify the expression. Use the substitution $$ w = 1 + u^{2} $$, so that $$ dw = 2u \, du  or$$ $$ u \, du = \frac{dw}{2} . $$Express $$ u^{4}  in $$terms of $$ w $$ and rewrite the integral accordingly. Then, integrate with respect to $$ w $$ using standard integration techniques for powers of $$ w .$$

Evaluate the integrals in Exercises 33–52.
∫ tan⁴(x) sec³(x) dx

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Key Concepts

Trigonometric Identities

Integration by Substitution

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