Problem 73 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The integral is \( \sin x + \frac{x}{2} + \frac{\sin(2x)}{4} + C \).

Step 1: Identify the Integration Components

The integral to solve is \( \int\left(\cos x+\cos ^{2} x\right) d x \). This is a sum of two functions: \( \cos x \) and \( \cos^2 x \). We will integrate each part separately and then combine the results.

Step 2: Integrate \( \cos x \)

The integral of \( \cos x \) is known to be \( \sin x + C \), where \( C \) is the constant of integration. This is straightforward based on basic trigonometric integrals.

Step 3: Transform \( \cos^2 x \) for Integration

The function \( \cos^2 x \) can be challenging to integrate directly. Use the trigonometric identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \) to transform the function. Then the integral becomes \( \int \frac{1 + \cos(2x)}{2} dx \).

Step 4: Integrate Transformed \( \cos^2 x \)

Now separate the integrals: \( \int \frac{1}{2} dx + \int \frac{\cos(2x)}{2} dx \).- The first integral is \( \frac{1}{2}x + C_1 \).- For the second, use the substitution method where \( u = 2x \), \( du = 2 dx \), resulting in \( \frac{1}{4} \sin(2x) + C_2 \).

Step 5: Combine Integration Results

Add the results from Steps 2 and 4:1. From \( \cos x \), we have \( \sin x \).2. From \( \cos^2 x \), combining gives: \( \frac{x}{2} + \frac{\sin(2x)}{4} \).Thus, the result of the entire integral is:\( \sin x + \frac{x}{2} + \frac{\sin(2x)}{4} + C \).

Step 6: State the Final Answer

The indefinite integral \( \int\left(\cos x+\cos^2 x\right) d x \) is given by:\[ \sin x + \frac{x}{2} + \frac{\sin(2x)}{4} + C \]where \( C \) is the constant of integration.

Key Concepts

Indefinite Integrals

In the realm of integral calculus, indefinite integrals are a fundamental concept. Essentially, an indefinite integral is a function that describes the set of all antiderivatives of a given function. Unlike definite integrals, indefinite integrals do not have limits of integration; instead, they include a constant of integration, typically denoted as \( C \). This constant is crucial because when taking the derivative of any antiderivative function, this constant disappears, leaving the original function. Considering the exercise \( \int(\cos x + \cos^2 x) \, dx \), each part (\( \cos x \) and \( \cos^2 x \)) is integrated separately. The result of these indefinite integrals provides solutions that include an undefined constant, symbolizing the family of solutions present with such integrations.

Trigonometric Identities

Trigonometric identities are tools that make solving integrals involving trigonometric functions much easier. For functions like \( \cos^2 x \), a direct integration isn’t straightforward. This is where trigonometric identities come to the rescue. One of the most useful identities in integral calculus is the double angle formula: \( \cos^2 x = \frac{1 + \cos(2x)}{2} \). This identity transforms a squared trigonometric function into a more manageable form to integrate. The use of such identities often simplifies complex trigonometric expressions, allowing us to break them down into simpler, easily integrable parts. In the given problem, transforming \( \cos^2 x \) using this identity allows the problem to be split into simpler integral parts, making it possible to find the antiderivative.

Substitution Method

The substitution method, often referred to as "u-substitution," is a powerful technique in calculus for simplifying the integration of complex functions. It works by substituting part of the integral with a new variable, simplifying the expression. Consider the integral \( \int \frac{\cos(2x)}{2} \, dx \). By setting \( u = 2x \), and recognizing that \( du = 2 \, dx \), we can substitute to find a much simpler form: \( \int \frac{1}{2} \cdot \frac{1}{2} \sin(u) \, du \), which integrates to \( \frac{1}{4} \sin(2x) \). The substitution method not only simplifies integration but also helps with reversing this process for antiderivatives in definite integrals. Through our example exercise, this method is used to handle parts of the integral that present complexities, ultimately assisting in the integration of \( \cos^2 x \) after applying trigonometric identities.