Problem 36 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=\cos ^{2} x-\sin ^{2} x $$

Step-by-Step Solution

Answer

The general antiderivative is \(F(x) = \frac{1}{2} \sin(2x) + C\).

Step 1: Identifying the Trigonometric Identity

Start by recognizing the trigonometric identity associated with the given function. The expression \(rac{1}{2}(\cos 2x)\) is relevant because \(\cos^2 x - \sin^2 x = \cos(2x)\).

Step 2: Use the Trigonometric Identity

Rewrite the function using the identified trigonometric identity: \(f(x) = \cos 2x\).

Step 3: Integrating the Function

Now, integrate \(\cos (2x)\) with respect to \(x\). Recall that the integral of \(\cos(ax)\) is \(\frac{1}{a}\sin(ax) + C\).

Step 4: Applying the Integral Formula

Apply the formula for integrating \(\cos(2x)\):\[\int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C\].

Step 5: Writing the General Antiderivative

The general antiderivative of the given function is therefore:\[F(x) = \frac{1}{2} \sin(2x) + C\].

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles. They are very useful in simplifying expressions and solving equations. In the context of our exercise, the identity \(cos^2 x - sin^2 x = cos(2x)\) helps simplify the function into a more integrable form.
This identity is a specific case of the double angle formula for cosine, which states that \(cos(2x) = sin^2 x - cos^2 x\). Recognizing these identities quickly is crucial since they simplify complex trigonometric expressions into simpler forms, facilitating easier integration or differentiation.
Understanding which identity to use comes with practice, as each trigonometric identity is applicable in different scenarios depending on the problem faced. Their primary goal is to make computation more straightforward and elegant.

Antiderivatives

Antiderivatives, also known as indefinite integrals, represent the reverse process of differentiation. When you find the antiderivative of a function \( f(x) \), you are essentially finding a function \( F(x) \) such that \( F'(x) = f(x) \).
In the context of the problem, we have the function \( f(x) = cos 2x \). To find its antiderivative, we seek a function \( F(x) \) whose derivative is \( cos 2x \).
Finding antiderivatives involves using known integration formulas or techniques, recognizing patterns, or substituting variables from given identities. The resulting general antiderivative includes an additional constant \( C \), which indicates that there are infinitely many antiderivatives due to the constant nature of integration.

Integration Techniques

Integration techniques are methods used to calculate the integral of a function. In our exercise, we use the direct integration technique followed by the application of a trigonometric identity to simplify the problem.
For the specific integration of \( cos(2x) \), we apply the formula: \[int cos(ax) \, dx = \frac{1}{a} sin(ax) + C \]
This formula emerges from a basic understanding of derivatives of trigonometric functions and scaling effects on the angle within the trigonometric function. It allows us to integrate functions where the angle is multiplied by a coefficient.
Another important technique is substitution, which isn’t directly used here but serves as a powerful tool in transforming an integral into an easier form.** Ultimately, a solid understanding of these techniques, combined with trigonometric identities, allows for calculating integrals of increasingly complex functions.