1. State the half-angle identities used to integrate sin²x and cos²x.

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The half-angle identities are trigonometric formulas that simplify expressions involving squared sine and cosine functions. They are derived from the double-angle identities.

The half-angle identity for sin²x is:

\sin^{2} x = \frac{1 - \cos 2 x}{2}

. This identity expresses sin²x in terms of cos(2x), making integration easier.

The half-angle identity for cos²x is:

\cos^{2} x = \frac{1 + \cos 2 x}{2}

. This identity expresses cos²x in terms of cos(2x), simplifying integration.

To integrate sin²x or cos²x, substitute the respective half-angle identity into the integral. For example, replace sin²x with

\frac{1 - \cos 2 x}{2}

After substitution, simplify the integral and proceed with standard integration techniques, such as integrating cos(2x) using the formula

\frac{\sin}{2 x}

divided by 2.

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Key Concepts

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

Half-Angle Identities

Half-angle identities are trigonometric identities that express the sine and cosine of half an angle in terms of the sine and cosine of the full angle. They are derived from the double angle formulas and are particularly useful in integration and simplifying trigonometric expressions. The identities are: sin²(x/2) = (1 - cos(x))/2 and cos²(x/2) = (1 + cos(x))/2.

Integration of Trigonometric Functions

Integrating trigonometric functions often requires the use of identities to simplify the integrand. For sin²x and cos²x, applying the half-angle identities allows us to rewrite these functions in a more manageable form, facilitating the integration process. This technique is essential for solving integrals that involve squared trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are fundamental tools in calculus for simplifying expressions and solving equations. Understanding these identities, including Pythagorean, angle sum, and half-angle identities, is crucial for effectively manipulating and integrating trigonometric functions.

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