Problem 61 - Applied Calculus

Question

Find the indefinite integrals. $$ \int(10+8 \sin (2 x)) d x $$

Step-by-Step Solution

Answer

The integral is \( 10x - 4\cos(2x) + C \).

Step 1: Break Down the Integral

The given integral is \( \int (10 + 8 \sin(2x)) \, dx \). Since integration is a linear operation, we can separate the integral into two parts: \( \int 10 \, dx \) and \( \int 8 \sin(2x) \, dx \).

Step 2: Integrate the Constant Term

Integrate the constant \(10\) with respect to \(x\). The integral of a constant \(a\) is \(ax + C\), where \(C\) is the integration constant. Therefore, \(\int 10 \, dx = 10x + C_1\).

Step 3: Integrate the Trigonometric Term

Integrate the term \(8 \sin(2x)\). To do this, use the substitution method. Let \(u = 2x\), which implies \(du = 2 \, dx\) or \(dx = \frac{du}{2}\). Substitute into the integral: \[ \int 8 \sin(2x) \, dx = \int 8 \sin(u) \cdot \frac{du}{2} = 4 \int \sin(u) \, du. \]

Step 4: Solve the Integral of the Substitution

The integral \(\int \sin(u) \, du\) is \(-\cos(u) + C\). Thus, \(4 \int \sin(u) \, du = -4 \cos(u) + C_2\).

Step 5: Substitute Back and Combine the Integrals

Replace \(u = 2x\) into the result from Step 4 to get \(-4\cos(2x) + C_2\). Now, combine the results from Step 2 and Step 4: \[ 10x - 4\cos(2x) + C, \] where \(C = C_1 + C_2\) is the new constant of integration.

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus, where we find the antiderivative of a given function. It's like the reverse of differentiation. There are various techniques used to solve integrals, and choosing the right one depends on the form of the function you're dealing with. Here are some common integration techniques:

• Basic Integration: This involves using fundamental formulas and rules to directly integrate functions. For instance, integrating constants or simple polynomial terms.
• Integration by Substitution: This method is used when an integral contains a composite function. By substituting parts of the integrand, we can simplify it into a more manageable form.
• Integration by Parts: Useful for products of functions, it is based on the product rule of differentiation and allows us to break down integrals into simpler parts.
• Partial Fraction Decomposition: Used when dealing with rational functions, this technique involves breaking down a complex fraction into simpler fractions that can be integrated individually.
• Trigonometric Integrals: Involves integrating functions containing trigonometric functions. Special identities and substitutions are often used to simplify the process.

In the exercise at hand, we mainly use basic integration and substitution techniques to tackle the problem efficiently.

Substitution Method

The substitution method is a powerful tool for solving complex integrals by transforming them into simpler ones. In essence, it "changes variables" to make integration easier. Here's how it works:

• Identify a part of the integral that can be substituted with a new variable. For example, if you have an expression like \(2x\), you set \(u = 2x\).
• Express \(dx\) in terms of \(du\). In our case, since \(du = 2 \, dx\), we find that \(dx = \frac{du}{2}\).
• Substitute these new variables into the integral to transform it. This typically simplifies the integral, making it easier to solve.
• Once the integration is complete, substitute back the original variable expressions to get the solution in terms of the original variable.

In the given problem, substitution is crucial for integrating \(8 \sin(2x)\), as it's rewritten in terms of \(u\) to simplify the process. This method is especially useful when dealing with trigonometric functions within the integral.

Trigonometric Functions

Trigonometric functions often appear in calculus problems, and knowing how to integrate them is essential. Common trigonometric functions include \(\sin(x)\), \(\cos(x)\), \(\tan(x)\), and their inverses and reciprocals. Here's what you should know about their integration:

• The integral of \(\sin(x)\) is \(-\cos(x) + C\), while the integral of \(\cos(x)\) is \(\sin(x) + C\).
• For \(\tan(x)\) and \(\sec(x)\), integration can be done using identities or transformations, like factoring the integral or using derivatives of specific functions.
• Trigonometric identities can be used to transform and simplify integrals involving these functions. For example, identities like \(\sin^2(x) + \cos^2(x) = 1\) can help simplify expressions.
• Substitution is frequently used with trigonometric functions to make integration manageable, as seen in the given problem with \(\sin(2x)\).

In our example, \(\sin(2x)\) is a key part of the problem. With the substitution method, we turn it into a simple \(\sin(u)\), which is much easier to integrate. Understanding these functions and techniques is crucial for handling any trigonometric integration problem effectively.