Problem 74 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The integral is \(-\cos x - \frac{x}{2} + \frac{1}{4} \sin(2x) + C\).

Step 1: Break Down the Integral

We start with the integral \( \int(\sin x - \sin^2 x) \, dx \). Notice that it is a combination of two parts: \( \int \sin x \, dx \) and \( \int \sin^2 x \, dx \). We will address each part separately.

Step 2: Integrate the First Term \( \int \sin x \, dx \)

The integral of \( \sin x \) is well-known: \( \int \sin x \, dx = -\cos x + C_1 \), where \( C_1 \) is a constant of integration.

Step 3: Use Trigonometric Identity for \( \sin^2 x \)

To integrate \( \sin^2 x \), we use the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \). Thus, \( \int \sin^2 x \, dx = \int \frac{1 - \cos(2x)}{2} \, dx \).

Step 4: Integrate \( \frac{1 - \cos(2x)}{2} \)

This integral can be split into two separate integrals: \( \int \frac{1}{2} \, dx - \int \frac{\cos(2x)}{2} \, dx \). The integral of \( \frac{1}{2} \, dx \) is \( \frac{x}{2} + C_2 \). For \( \int \frac{\cos(2x)}{2} \, dx \), use substitution, letting \( u = 2x \), then \( du = 2 \, dx \), or \( dx = \frac{du}{2} \), yielding \( \int \frac{1}{2} \cos(u) \, \frac{du}{2} = \frac{1}{4} \sin(u) + C_3 = \frac{1}{4} \sin(2x) + C_3 \).

Step 5: Combine and Simplify

Summing up the integrals, we get \( \int \frac{1 - \cos(2x)}{2} \, dx = \frac{x}{2} - \frac{1}{4} \sin(2x) + C_2 + C_3 \).

Step 6: Construct the General Solution

Now combine all parts: \( \int (\sin x - \sin^2 x) \, dx = \int \sin x \, dx - \int \sin^2 x \, dx = (-\cos x) - \left(\frac{x}{2} - \frac{1}{4} \sin(2x)\right) + C \). Simplify to get: \(-\cos x - \frac{x}{2} + \frac{1}{4}\sin(2x) + C \), where \( C \) is the constant of integration.

Key Concepts

Trigonometric Identities in Integration

Trigonometric identities are powerful tools in calculus that help simplify and solve integrals involving trigonometric functions. These identities allow us to transform a given expression into a more manageable form. In this exercise, we used a well-known trigonometric identity to simplify the integral of \(\sin^2 x\). This identity is:
\[\sin^2 x = \frac{1 - \cos(2x)}{2}.\]
This transformation is crucial because it simplifies the original trigonometric expression into a form that is easier to integrate. When faced with integrals of squared trigonometric functions, applying such identities often results in simpler expressions, making it easier to perform the integration step-by-step. Identifying and using appropriate trigonometric identities is a skill that will enhance your ability to solve complex integrals effectively.

Integration by Substitution

Integration by substitution is a method used to simplify the process of integration when dealing with complicated functions. It works by replacing a part of the integral with a new variable, typically denoted as \(u\). This process is analogous to the reverse of the chain rule in differentiation. In this exercise, substitution was key when addressing the integral involving \(\cos(2x)\). When we set \(u = 2x\), it followed that \(du = 2 \, dx\), which allows us to rewrite the differential part as \(dx = \frac{du}{2}\).

• This transformation turns the original integral into a simpler form, allowing us to integrate with respect to \(u\) instead of the original variable \(x\).
• Subsequently, solving \( \int \frac{\cos(2x)}{2} \, dx \) becomes \( \int \frac{1}{4} \cos(u) \, du \).
• This results in the integral of \(\cos(u)\), which is straightforward to integrate as \(\frac{1}{4} \sin(u)\), or equivalently \(\frac{1}{4} \sin(2x)\).

Substitution is a versatile and powerful technique, especially when integrals include composite functions. Mastery of this concept simplifies solving a broad range of calculus problems.

Understanding the Constants of Integration

When calculating indefinite integrals, the result includes a constant term known as the constant of integration, represented by \(C\). This constant accounts for the fact that differentiation of constant terms results in zero, which means any anti-derivative might vary by just a constant value. In our integration steps, different constants arose, such as \(C_1, C_2,\) and \(C_3\). In the final solution, these are typically combined into a single constant \(C\).

• The process illustrates that there can be multiple ways to express the same indefinite integral, each differing merely by a constant.
• This constant signifies the infinite set of parallel solutions we can have for an indefinite integral.
• For instance, different initial conditions in a physical problem might lead to solutions that differ by constant values. Hence, observing the constants of integration is crucial in the wider context of applying solutions to real-world scenarios.

Understanding and properly accounting for these constants is vital. It ensures we recognize the general solution includes potentially myriad specific solutions, each fitting various initial conditions or contexts.