Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int 4(4 x+3)^{4} d x$$

Step-by-Step Solution

Answer

The short answer is \(\int 4(4 x+3)^{4} d x = \frac{(4x + 3)^5}{5} + C\).

Step 1: Choose the substitution

Let \(u = 4x + 3\). In this case, we selected this substitution because the derivative of \(u = 4x + 3\) is simply a constant which will allow for easy integration.

Step 2: Find the derivative of the substitution

Compute the derivative of \(u = 4x + 3\) with respect to \(x\): \[\frac{d u}{d x} = 4\] Now find \(dx\) in terms of \(du\): \[dx = \frac{1}{4} du\]

Step 3: Replace dx and substitute u in the integral

Replace \(4x + 3\) with \(u\) and replace \(dx\) with \(\frac{1}{4}du\): \[\int 4(4 x+3)^{4} d x = \int 4 u^4 (\frac{1}{4} du)\]

Step 4: Simplify and integrate

Simplify the integral and after that, integrate with respect to \(u\): \[\int u^4 du\] Now apply the Power Rule of Integration, which states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) where \(C\) is the constant of integration: \[\frac{u^{5}}{5} + C\]

Step 5: Replace u with the original expression

Replace \(u\) by its original expression in terms of \(x\): \[\frac{(4x + 3)^5}{5} + C\] The indefinite integral of the given function is: \[\int 4(4 x+3)^{4} d x = \frac{(4x + 3)^5}{5} + C\]

Key Concepts

Substitution Method

The substitution method is a popular integration technique. It simplifies complex integrals by transforming them into simpler ones. Here's a step-by-step understanding of how it works, using our original exercise.When encountering an integral like \( \int 4(4x+3)^4 \, dx \), it's often beneficial to identify a part of the integrand as a new variable. In this exercise, we select \( u = 4x + 3 \). This substitution helps because the derivative, \( \frac{du}{dx} = 4 \), is a constant, making calculations easier. After substituting, we express \( dx \) in terms of \( du \) as \( dx = \frac{1}{4} du \). Rewriting the integral with these terms leads to \( \int 4u^4 (\frac{1}{4} du) \), simplifying further to \( \int u^4 du \). The substitution reduces the original complexity and sets the stage for integration using simpler functions.

Power Rule of Integration

The Power Rule of Integration is one of the most essential rules for finding antiderivatives. It helps integrate functions of the form \( x^n \). In our example, after using the substitution method, we arrive at \( \int u^4 du \). This is where the Power Rule comes into play. The Power Rule states:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

where \( n eq -1 \). The constant \( C \), known as the constant of integration, accounts for any constant that could have been differentiated to zero.Applying this rule to \( \int u^4 du \), results in the antiderivative \( \frac{u^5}{5} + C \). This step is crucial as it transitions from performing algebraic manipulations to achieving the solution.

Integration Techniques

There are several techniques in calculus developed to tackle different types of integrals. The choice of technique largely depends on the form of the function being integrated. For our current problem, the combination of the Substitution Method and the Power Rule serves as the appropriate approach. Understanding the following common integration techniques can expand problem-solving skills:

• **Substitution Method:** Used when an integral contains a function and its derivative. Transform the integral into something simpler.
• **Integration by Parts:** Useful for integrating the product of two functions. It is based on the product rule for differentiation.
• **Partial Fraction Decomposition:** Applies to rational functions, splitting them into simpler fractions that are easier to integrate.
• **Trigonometric Integrals:** Used when the integrand involves trigonometric functions.

Each technique has its set of rules and best-use scenarios. Familiarity with these techniques enhances efficiency in solving indefinite integrals.