Problem 72 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int\left(\cos ^{2} x-\sin ^{2} x\right) d x $$

Step-by-Step Solution

Answer

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

Step 1: Recognize Trigonometric Identity

Notice the expression inside the integral, \(\cos^2 x - \sin^2 x\). Recognize that this follows the trigonometric identity known as the double angle formula: \(\cos(2x) = \cos^2 x - \sin^2 x\). This simplifies the integral \(\int (\cos^2 x - \sin^2 x) dx\) to \(\int \cos(2x) \, dx\).

Step 2: Integration of Cosine Function

Integrate \(\cos(2x)\) using a basic integration formula. Recall that \(\int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C\). In this case, \(k = 2\), so \(\int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C\), where \(C\) is the constant of integration.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express relationships between trigonometric functions like sine, cosine, and tangent. These identities are fundamental in simplifying complex trigonometric expressions, making integration more manageable. In trigonometry, common identities include:

• Pythagorean identities, such as \[\sin^2 x + \cos^2 x = 1\]
• Sum and difference formulas, \[\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\]
• Double angle formulas, which we'll delve into next.

Identities simplify the process of integrating trigonometric functions. By recognizing these relationships, complex transformations in integrals become simpler tasks, easing your calculus journey.

Double Angle Formula

A double angle formula is a specific trigonometric identity used to simplify the integration of functions like the one in our problem. The double angle formula for cosine is:\[\cos(2x) = \cos^2 x - \sin^2 x\]This particular identity is derived by using sum formulas and is pivotal in transforming multiple angle expressions.
In the context of integrating \((\cos^2 x - \sin^2 x)\), we can replace it with \(\cos(2x)\). This transition creates a more straightforward path to solving the integral. Understanding and applying such transformations reduces integration to base forms and is an invaluable skill for solving complex trigonometric integrals.

Integration of Trigonometric Functions

The integration of trigonometric functions is a foundational skill in calculus. To integrate functions involving trigonometric terms, knowing the base formulas is crucial.
For instance, to integrate \( \cos(kx) \), use the formula:\[\int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C\]where \(C\) is the constant of integration.
In our problem, we've simplified \((\cos^2 x - \sin^2 x)\) to \(\cos(2x)\), allowing for the application of this formula. Plugging in \(k = 2\), we get:\[\int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C\]By mastering these integral expressions, students can tackle many complex calculus problems confidently and effectively.