Question

Use the double-angle formulas to evaluate the following integrals. \(\int \sin ^{2} x d x+\int \cos ^{2} x d x\)

Short answer

The value of the integral is \( x + C \).

Step-by-step solution

Recall the Double-Angle Identity

Recall that the double-angle identity for cosine states: \( \cos(2x) = \cos^2(x) - \sin^2(x) \). Also, it can be written in terms of sine and cosine as \( \cos(2x) = 2\cos^2(x) - 1 \) or \( \cos(2x) = 1 - 2\sin^2(x) \).

Express Trigonometric Squares Using Double-Angle Formula

Use the identities to express \( \sin^2(x) \) and \( \cos^2(x) \): \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) and \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \).

Rewrite the Integral Using Identities

Rewrite the integral using the identities: \[ \int \sin^2(x)\,dx + \int \cos^2(x)\,dx = \int \frac{1 - \cos(2x)}{2}\,dx + \int \frac{1 + \cos(2x)}{2}\,dx. \]

Simplify the Expression

Combine the integrals:\[ \int \left(\frac{1 - \cos(2x)}{2} + \frac{1 + \cos(2x)}{2}\right)\,dx = \int \frac{1}{2} + \frac{1}{2}\,dx = \int 1\,dx. \]

Evaluate the Integral

The integral \( \int 1\,dx \) is simply \( x + C \), where \( C \) represents the constant of integration.

Key concepts

Trigonometric Integrals

When dealing with trigonometric integrals, it's often about integrating functions involving sine and cosine. In this specific problem, the task is to evaluate the integral of the square of sine and cosine, which are common expressions in calculus.

Key to solving such integrals is identifying when and how to apply identities and simplifications. Sine squared and cosine squared are tricky because they are not directly integrable. Therefore, we turn to known trigonometric identities, like the Double-Angle Formulas, to assist us.

• The purpose of simplifying these trigonometric squares is to rewrite them so that the integration becomes manageable.
• This often involves expressing these functions in terms that are easier to handle analytically.

Utilizing such techniques reduces the complexity of the integration process and helps bridge the gap between simple and complex trigonometric integrals.

Sine and Cosine Identities

Sine and Cosine Identities, particularly the Double-Angle Formulas, are vital tools in solving integrals involving trigonometric functions. The main formula used here is:

• \[\cos(2x) = \cos^2(x) - \sin^2(x)\right. \\]
• Also, it can be rewritten as either \[\cos(2x) = 2\cos^2(x) - 1\right.\\] or \[\cos(2x) = 1 - 2\sin^2(x)\right.\\] depending on what form is needed.
• These identities can be solved to define the squares: \[\sin^2(x) = \frac{1 - \cos(2x)}{2}\right.\\] and \[\cos^2(x) = \frac{1 + \cos(2x)}{2}\right.\\]

Upon using these identities, the calculation of integrals becomes straightforward because they convert the trigonometric squares into terms involving a linear cosine function, which can be integrated seamlessly. Such transformations are essential for simplifying the calculus process and turning complex problems into solvable ones.

Integration Techniques

In this exercise, several integration techniques are seamlessly blended to solve the integral effectively. Breaking down the sophisticated problem involved a few systematic techniques.

• The transformation of trigonometric squares into forms involving cosine using double-angle identities.
• Utilization of fundamental rules of addition and integration to collapse complex terms into simpler forms.
• Carry out elementary integration of straightforward terms, which in this case involved recognizing that the integral of a constant like "1" over "dx" results in "x".

By following these steps, the process of integrating complicated trigonometric expressions becomes more methodical, emphasizing the importance of mathematical identities and strategic methods.
Such problem-solving tactics are crucial in calculus, particularly when handling trigonometric integrals, turning challenging problems into solvable steps with ease.