Problem 21 - Calculus

Question

Use the method of substitution to find each of the following indefinite integrals. $$ \int x \sqrt{x^{2}+4} d x $$

Step-by-Step Solution

Answer

The solution is \( \frac{1}{3} (x^2 + 4)^{3/2} + C \).

Step 1: Identify the Substitution

To apply substitution, first identify a part of the integral that can be replaced with a single variable to simplify the calculation process. In this integral, the expression inside the square root, \(x^2 + 4\), is a suitable candidate for substitution.

Step 2: Define the Substitution

Let \( u = x^2 + 4 \). Differentiate both sides with respect to \( x \) to find \( du \). This gives us \( du = 2x \, dx \).

Step 3: Solve for dx

Rearrange the expression \( du = 2x \, dx \) to express \( dx \) in terms of \( du \) and \( x \). We get \( dx = \frac{du}{2x} \).

Step 4: Express the Integral in terms of u

Substitute \( u = x^2 + 4 \) and \( dx = \frac{du}{2x} \) into the integral: \[ \int x \sqrt{u} \frac{du}{2x} = \int \frac{1}{2} \sqrt{u} \, du \].

Step 5: Integrate with Respect to u

Now, integrate \( \int \frac{1}{2} u^{1/2} \, du \). The antiderivative of \( u^{1/2} \) is \( \frac{2}{3} u^{3/2} \), so:\[ \int \frac{1}{2} \sqrt{u} \, du = \frac{1}{2} \cdot \frac{2}{3} u^{3/2} = \frac{1}{3} u^{3/2} + C \] where \( C \) is the constant of integration.

Step 6: Back-substitute for x

Replace \( u \) with \( x^2 + 4 \) to express the integral back in terms of \( x \). Thus,\[ \int x \sqrt{x^2 + 4} \, dx = \frac{1}{3} (x^2 + 4)^{3/2} + C \].

Key Concepts

Understanding Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus. They represent the collection of all antiderivatives of a given function. When you hear the term "indefinite integral," think of it as the reverse process of differentiation. An indefinite integral does not have upper or lower limits, and therefore, its result is broad and generalized, typically including an arbitrary constant called the constant of integration.

Here's why the constant is crucial: when you differentiate a constant, you get zero. So, adding or subtracting any constant does not affect the differentiation result. This means there are infinite antiderivatives for any function, differing by just a constant. Understanding indefinite integrals is thus key in solving a wide range of problems in calculus, allowing for the determination of the original function given its derivative.

In essence, solving indefinite integrals will help you find functions that could model real-world situations, where you often have the rate of change and need the original quantity.

Mastering the Substitution Method

The substitution method is a flexible tool in calculus for evaluating integrals, particularly those that look complicated at first glance.

This technique involves substituting a part of the integrand with a single variable, simplifying the integration process. Here's how it works:

• Choose a substitution: Look for a part of the integral that looks cumbersome or is nested within other functions. In the solution given, the substitution involves setting \( u = x^2 + 4 \).
• Find the differential: Once you choose \( u \), differentiate it to find \( du \). For example, if \( u = x^2 + 4 \), then \( du = 2x \, dx \).
• Substitute in the integral: Change all expressions in the integral from \( x \) to \( u \) and adjust the differential accordingly. This transforms the integral into a simpler form, rendering it easier to solve.

The substitution method is powerful because it turns complex integrals into manageable pieces. Practice will help you identify substitution candidates more efficiently, making this approach second nature.

Defining the Antiderivative

An antiderivative is simply the function you need to differentiate to obtain the original given function. When you're calculating an indefinite integral, you're essentially searching for an antiderivative.

In calculus, finding an antiderivative means reversing differentiation. For instance, if the derivative of a function \( f(x) \) is \( f'(x) \), then \( f(x) \) is an antiderivative of \( f'(x) \).

• General Formula: For a simple power function \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} \), as long as \( n eq -1 \).
• Adding Constant: Remember that integrating adds a constant \( C \), representing all possible antiderivatives.

This concept is vital for both theoretical analysis and practical applications such as calculating areas under curves or solving differential equations.