Question

Evaluate the integrals in Exercises 33–52.∫ sec⁶(x) dx

Accepted Answer

Recognize that the integral involves a high even power of secant: $$\int \sec$$^{6}$$(x) \, dx. To $$simplify, express $$\sec$$^{6}$$(x) as$$ $$\sec$$^{4}$$(x) \cdot \sec$$^{2}$$(x). $$Rewrite $$\sec$$^{4}$$(x) as$$ $$(\sec$$^{2}$$(x))^{2}$$, so the integral becomes $$\int (\sec$$^{2}$$(x))^{2}$$ \cdot \$$sec^{2}(x$$) \, dx = \int \$$sec^{4}(x$$) \cdot \$$sec^{2}(x$$) \, dx$$. Use the identity $$\$$sec^{2}(x) = 1$$ + \$$tan^{2}(x$$)$$ to express powers of secant in terms of tangent, or alternatively, use the reduction formula for powers of secant: $$\int \$$sec^{n}(x$$) \, dx = \frac{\$$sec^{n-2}(x$$) 	an(x)}{n-1} + \frac{n-2}{n-1} \int \$$sec^{n-2}(x$$) \, dx$$ for n$$ \u003e 1$$. Apply the reduction formula with n=6$ to reduce the integral $$\int \$$sec^{6}(x$$) \, dx$$ to an expression involving $$\int \$$sec^{4}(x$$) \, dx$$. Repeat the reduction process on $$\int \$$sec^{4}(x$$) \, dx$$ until you reach integrals of $$\$$sec^{2}(x$$)$$, which is straightforward to integrate, and then combine all parts to express the original integral.

Evaluate the integrals in Exercises 33–52.
∫ sec⁶(x) dx

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Key Concepts

Integration of Powers of Secant

Trigonometric Identities

Substitution Method