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

Verified step by step guidance

1. 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.
2. 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 \(\cos^2(y) = \frac{1 + \cos(2y)}{2}\).
3. 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.
4. 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.
5. 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.

Key Concepts

Trigonometric Identities

Trigonometric identities, such as power-reduction and double-angle formulas, simplify expressions involving powers of sine and cosine. For example, sin²(y) and cos²(y) can be rewritten using identities to make integration more manageable.

Definite Integration

Definite integration calculates the exact area under a curve between two limits. Here, the integral from 0 to π requires evaluating the antiderivative at these bounds and subtracting to find the integral's value.

Integration of Powers of Sine and Cosine

Integrating powers of sine and cosine often involves using reduction formulas or rewriting powers in terms of multiple angles. This approach transforms complex integrals into sums of simpler trigonometric integrals.