Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx

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Recognize that the integral involves the square of the sine function: \(\int_0^{\frac{\pi}{2}} \sin^2(x) \, dx\).

Use the power-reduction identity to rewrite \(\sin^2(x)\) in a more integrable form: \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\).

Substitute this identity into the integral to get \(\int_0^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} \, dx\).

Split the integral into two separate integrals: \(\frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx - \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(2x) \, dx\).

Evaluate each integral separately using basic integration rules, then combine the results to express the value of the original integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating a definite integral 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 integrals like ∫ sin²(x) dx, using identities such as the power-reduction formula sin²(x) = (1 - cos(2x))/2 simplifies the integral into a more manageable form.

Fundamental Theorem of Calculus

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

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