Evaluate the integrals in Exercises 53–58.
∫ from 0 to π/2 of sin(x) cos(x) dx

Verified step by step guidance

1. Recognize that the integral is \( \int_0^{\frac{\pi}{2}} \sin(x) \cos(x) \, dx \). This is a definite integral with limits from 0 to \( \frac{\pi}{2} \).
2. Use a trigonometric identity or substitution to simplify the integrand. One useful identity is \( \sin(2x) = 2 \sin(x) \cos(x) \), so \( \sin(x) \cos(x) = \frac{1}{2} \sin(2x) \).
3. Rewrite the integral using this identity: \( \int_0^{\frac{\pi}{2}} \sin(x) \cos(x) \, dx = \int_0^{\frac{\pi}{2}} \frac{1}{2} \sin(2x) \, dx \).
4. Factor out the constant \( \frac{1}{2} \) from the integral: \( \frac{1}{2} \int_0^{\frac{\pi}{2}} \sin(2x) \, dx \).
5. Perform a substitution to integrate \( \sin(2x) \): let \( u = 2x \), then \( du = 2 dx \) or \( dx = \frac{du}{2} \). Change the limits accordingly and integrate \( \sin(u) \) with respect to \( u \).

Key Concepts

Definite Integrals

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. For example, the product-to-sum identity sin(x)cos(x) = (1/2)sin(2x) simplifies integration by converting products into sums, making the integral easier to evaluate.

Fundamental Theorem of Calculus

This theorem links differentiation and integration, stating that if F is an antiderivative of f on [a, b], then ∫ from a to b of f(x) dx = F(b) - F(a). It allows us to evaluate definite integrals by finding antiderivatives and computing their difference at the bounds.