Problem 36 - Calculus Single Variable

Question

Use trigonometric identities to compute the indefinite integrals. $$ \int\left(\cos ^{2}(x)-\sin ^{2}(x)\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{1}{2} \sin(2x) + C \).

Step 1: Identify the Integral Expression

The integral to be solved is \( \int (\cos^2(x) - \sin^2(x)) \, dx \). We need to find an antiderivative for this expression.

Step 2: Recall Trigonometric Identity

Remember the trigonometric identity: \( \cos(2x) = \cos^2(x) - \sin^2(x) \). This identity matches the form of the integrand.

Step 3: Convert Integral Using Identity

Using the identity from Step 2, replace \( \cos^2(x) - \sin^2(x) \) with \( \cos(2x) \). The integral becomes \( \int \cos(2x) \, dx \).

Step 4: Integrate the New Expression

To integrate \( \cos(2x) \), use the rule \( \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \). Therefore, \( \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C \).

Step 5: Write Down the Final Result

Thus, the indefinite integral of \( \cos^2(x) - \sin^2(x) \) is \( \frac{1}{2} \sin(2x) + C \), where \( C \) is the constant of integration.

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools that simplify complex mathematical expressions, especially in integration and calculus. They help us convert trigonometric functions into other forms that may be easier to integrate or differentiate. One important identity we encounter in calculus is the double-angle identity for cosine:

• \( \cos(2x) = \cos^2(x) - \sin^2(x) \)

This identity allows us to transform the difference of squares of cosine and sine into a single cosine term with a doubled angle. In the context of our integral, using this identity makes evaluating the integral much simpler.
It is always helpful to remember the common trigonometric identities like Pythagorean identities, angle sum formulas, and double angle formulas. These can be applied in various calculus problems to bridge gaps between trigonometric expressions and their integrals or derivatives.

Antiderivatives

The concept of antiderivatives, often referred to as indefinite integrals, involves finding a function whose derivative is the given function. In other words, if you have a function where you know its derivative, the antiderivative helps you recover the original function up to a constant multiple. This process is the reverse of differentiation.
For example, when we found the antiderivative of \( \cos(2x) \), we used the rule:

• \( \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \)

In our solution, \( k \) is 2, leading to \( \frac{1}{2} \sin(2x) + C \) as our antiderivative. Here, \( C \) represents the constant of integration, reminding us that antiderivatives are not unique and differ by a constant. Antiderivatives are crucial because they are foundational for solving indefinite integrals and allow us to move from derivatives back to functions.

Integration Techniques

Integration techniques are methods used to find the integrals of functions, often requiring converting complex expressions into simple forms that are easier to integrate. Among these techniques, substitution and leveraging trigonometric identities can make integration straightforward.
In the exercise, after identifying the integral \( \int (\cos^2(x) - \sin^2(x)) \, dx \), the clever use of the trigonometric identity:

• \( \cos^2(x) - \sin^2(x) = \cos(2x) \)

allowed us to express the integral in a more workable form: \( \int \cos(2x) \, dx \). With this transformed integral, the calculation became straightforward using the formula for the integral of cosine of a multiple angle.
Choosing the right technique can significantly simplify calculations by reducing complex expressions to forms that are solvable using basic integration formulas. This highlights the importance of familiarizing oneself with different identities and techniques that can be applied in various situations to solve integrals efficiently.