Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx

Verified step by step guidance

1. Recognize that the integral is \( \int_0^{\pi} 8 \cos^4(2\pi x) \, dx \). The constant 8 can be factored out of the integral, so rewrite it as \( 8 \int_0^{\pi} \cos^4(2\pi x) \, dx \).
2. Use the power-reduction formula to express \( \cos^4(\theta) \) in terms of cosines of multiple angles. Recall that \( \cos^4(\theta) = \left( \cos^2(\theta) \right)^2 \) and \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \).
3. Substitute \( \cos^2(2\pi x) = \frac{1 + \cos(4\pi x)}{2} \) into the expression and then square it to get \( \cos^4(2\pi x) = \left( \frac{1 + \cos(4\pi x)}{2} \right)^2 \).
4. Expand the squared term to get \( \cos^4(2\pi x) = \frac{1}{4} \left( 1 + 2\cos(4\pi x) + \cos^2(4\pi x) \right) \). Then apply the power-reduction formula again to \( \cos^2(4\pi x) \) to express it in terms of cosines of multiple angles.
5. Rewrite the integral as a sum of integrals of cosines with different frequencies, integrate each term over \( 0 \) to \( \pi \), and then multiply the result by 8. Remember to use the fact that \( \int_0^{\pi} \cos(kx) \, dx = 0 \) for nonzero integer multiples \( k \) of \( \pi \).

Key Concepts

Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Trigonometric Power Reduction

Power reduction formulas transform powers of trigonometric functions into expressions involving multiple angles with lower powers. For example, cos⁴(θ) can be rewritten using cos(2θ) and cos(4θ) terms, simplifying integration by converting powers into sums of cosines.

Integration of Cosine Functions

Integrating cosine functions involves using standard integral formulas, such as ∫cos(kx) dx = (1/k) sin(kx) + C. When combined with definite limits, evaluating these integrals requires substituting the limits into the antiderivative and calculating the difference.