Question

Evaluate the integrals in Exercises 1–22.∫₀^π 8 sin⁴(y) cos²(y) dy

Accepted Answer

Recognize that the integral is $$\$$int_0$$^{\pi} 8 \sin^4$(y) \cos^2$(y) \, dy. $$The integrand involves powers of sine and cosine, so consider using trigonometric identities to simplify the expression before integrating. Use the power-reduction identities to express $$\sin^4$(y)$$ and $$\cos^2$(y) in $$terms of cosines of multiple angles. For example, recall that $$\sin^2$(y) = \frac{1 - \cos(2y)}{2}$$ and $$\cos^2$(y) = \frac{1 + \cos(2y)}{2}. $$Rewrite $$\sin^4$(y) as$$ $$(\sin^2$(y))^2$$ = \left( \frac{1 - \cos(2y)}{2} \$$right)^2$$ and substitute this along with the expression for $$\cos^2$(y)$$ into the integrand. Expand the product and simplify the resulting expression into a sum of cosines with different arguments. This will transform the integral into a sum of integrals of cosines, which are easier to evaluate. Integrate each term separately over the interval $$[0, \pi]$$ using the integral formula $$\int \cos(k y) \, dy = \frac{\sin(k y)}{k}$, and then apply the limits of integration to find the value of the integral.

Evaluate the integrals in Exercises 1–22.
∫₀^π 8 sin⁴(y) cos²(y) dy

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Key Concepts

Trigonometric Identities

Definite Integration

Integration of Powers of Sine and Cosine