Problem 15 - Calculus

Question

Use the method of substitution to find each of the following indefinite integrals. $$ \int \sqrt{3 x+2} d x $$

Step-by-Step Solution

Answer

\( \frac{2}{9} (3x + 2)^{3/2} + C \)

Step 1: Identify the Inner Function

First, identify the inner function for substitution. Here, it is \( u = 3x + 2 \).

Step 2: Differentiate the Inner Function

Differentiate \( u = 3x + 2 \) to find \( du \). This gives \( \frac{du}{dx} = 3 \), so \( du = 3\, dx \).

Step 3: Solve for \( dx \) in Terms of \( du \)

From the equation \( du = 3\, dx \), solve for \( dx \) which gives \( dx = \frac{du}{3} \).

Step 4: Rewrite the Integral Using \( u \)

Substitute \( u = 3x + 2 \) and \( dx = \frac{du}{3} \) into the integral:\[\int \sqrt{u} \cdot \frac{du}{3}\] which simplifies to \[ \frac{1}{3} \int u^{1/2} \, du \].

Step 5: Integrate with Respect to \( u \)

Integrate \( \frac{1}{3} \int u^{1/2} \, du \). Use the power rule for integration to find:\[\frac{1}{3} \left( \frac{u^{3/2}}{3/2} \right) = \frac{2}{9} u^{3/2} + C\] where \( C \) is the integration constant.

Step 6: Substitute Back to \( x \)

Replace \( u \) back with \( 3x + 2 \) to express the final answer in terms of \( x \):\[\frac{2}{9} (3x + 2)^{3/2} + C\]

Key Concepts

Indefinite Integrals

Indefinite integrals are fundamental in calculus, capturing the concept of finding antiderivatives. Unlike definite integrals, which have specific upper and lower limits, indefinite integrals look for the general form of an antiderivative. The notation typically used for indefinite integrals is \( \int f(x)\, dx \), where \( f(x) \) is the function to be integrated, and \( dx \) indicates the variable of integration.
To integrate, we need to determine the original function whose derivative is \( f(x) \). Unlike derivatives that yield a precise slope, integrals give a family of functions. These functions vary by a constant value, known as the constant of integration, \( C \). This is because any constant will disappear in the process of differentiation, so we add \( C \) to encompass all possible solutions.

• Process of Integration: Involves reversing differentiation.
• Notion of Constant: The result always includes \( +C \).
• Independent of Limits: Describes general antiderivatives, without specific limits.

Integration Techniques

In calculus, integration techniques are methods developed to solve various integral problems effectively. One of the notable methods is the substitution method, also known as \( u \)-substitution. This method is extremely useful when the integrand involves a composite function, making direct integration complex. The basic idea behind substitution is to simplify the integral by changing variables, which can sometimes turn a complicated problem into a straightforward one.
The substitution process involves a few key steps:

• Identify the inner function: Start by spotting the part of the integral that can be substituted by a single variable, \( u \).
• Differentiate the inner function: Differentiate this \( u \) with respect to \( x \) to find \( du \), equating \( dx \).
• Replace the terms: Substitute \( u \) and the corresponding \( dx \) in the integral, transforming the original integral to a simpler form.
• Integrate with respect to \( u \): Solve the resulting integral in terms of \( u \) and then substitute back to the original variable.

This method is particularly advantageous in handling integrands that are derivatives of functions within them, as showcased in problems involving square roots and polynomial expressions. It's an indispensable technique for mastering integration.

Power Rule for Integration

The power rule for integration is one of the most commonly used techniques for finding indefinite integrals. It provides a straightforward approach to integrate functions that are in the form of \( x^n \), where \( n \) is not equal to \(-1\). This rule is essentially the reverse process of the power rule in differentiation.
According to the power rule for integration, the integral of \( x^n \) with respect to \( x \) is given by:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where \( n eq -1 \) and \( C \) is the constant of integration. This formula helps us add one to the power and divide by the new power, making it a handy tool in calculus for solving polynomials.

A few essential points about the power rule:

• Not applicable when \( n = -1 \), since it would lead to division by zero. In such cases, use the natural logarithm for integration.
• Useful for straightforward integration of polynomials, regardless of whether they're part of a larger expression.
• Fundamentally based on reversing the derivative, which is efficient for direct polynomial expressions.

The power rule effectively aids in reducing complex problems to simpler forms, making integration a more manageable task.