Question

Use reduction formulas to evaluate the integrals in Exercises 41–50.∫ 2 $$sin^2(t$$) $$sec^4(t) dt$$

Accepted Answer

Start by rewriting the integral to clearly identify the powers of trigonometric functions: $$\int 2 \sin$$^{2}$$(t) \sec$$^{4}$$(t) \, dt. $$Recall that $$\sec(t) = \frac{1}{\cos(t)}$$, so $$\sec$$^{4}$$(t) = \frac{1}{\cos$$^{4}$$(t)}. $$This allows rewriting the integral as $$\int 2 \sin$$^{2}$$(t) \frac{1}{\cos$$^{4}$$(t)} \, dt. $$Express $$\sin$$^{2}$$(t) in $$terms of $$\cos(t)$$ using the Pythagorean identity: $$\sin$$^{2}$$(t) = 1 - \cos$$^{2}$$(t). $$Substitute this into the integral to get $$\int 2 (1 - \cos$$^{2}$$(t)) \frac{1}{\cos$$^{4}$$(t)} \, dt. $$Split the integral into two separate integrals: $$\int 2 \frac{1}{\cos$$^{4}$$(t)} \, dt - \int 2 \frac{\cos$$^{2}$$(t)}{\cos$$^{4}$$(t)} \, dt$$, which simplifies to $$\int 2 \sec$$^{4}$$(t) \, dt - \int 2 \sec$$^{2}$$(t) \, dt. $$Use the reduction formulas for $$\int \sec$$^{n}$$(t) \, dt to $$evaluate each integral separately. The reduction formula for $$\int \sec$$^{n}$$(t) \, dt is$$: $$\int \sec$$^{n}$$(t) \, dt = \frac{\sec$$^{n-2}$$(t) 	an(t)}{n-1} + \frac{n-2}{n-1} \int \sec$$^{n-2}$$(t) \, dt$$ for $$n \u003e 1. $$Apply this formula to $$\int \sec$$^{4}$$(t) \, dt$$ and recall that $$\int \sec$$^{2}$$(t) \, dt = 	an(t) + C.$$

Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 2 sin^2(t) sec^4(t) dt

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Key Concepts

Reduction Formulas

Trigonometric Identities

Integration Techniques for Trigonometric Functions

Use reduction formulas to evaluate the integrals in Exercises 41–50.∫ 2 sin^2(t) sec^4(t) dt