Problem 23 - Calculus for Biology and Medicine

Question

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \frac{3 x}{1+2 x^{2}} d x $$

Step-by-Step Solution

Answer

\( \frac{3}{4} \ln|1 + 2x^2| + C \)

Step 1: Identify Substitution

Start by identifying a substitution to simplify the integral. Notice that the derivative of the denominator, \(1 + 2x^2\), is related to the numerator, \(3x\). Let's try the substitution \( u = 1 + 2x^2 \).

Step 2: Differentiate Substitution

Differentiate the chosen substitution: \( du = \frac{d}{dx}(1 + 2x^2) \cdot dx = 4x \cdot dx \). In order to express \( dx \) in terms of \( du \) and \( x \), divide both sides by 4: \( du = 4x \cdot dx \Rightarrow dx = \frac{1}{4x} du \).

Step 3: Substitute in Original Integral

Replace \(1 + 2x^2\) by \(u\) in the integral and express \(dx\) using \(du\):\[ \int \frac{3x}{1+2x^2} dx = \int \frac{3x}{u} \cdot \frac{1}{4x} du \]Here, the \(x\) terms cancel out: \[ = \int \frac{3}{4} \cdot \frac{1}{u} du \]

Step 4: Integrate Resulting Expression

The expression simplifies to a basic integral:\[ \int \frac{3}{4} \cdot \frac{1}{u} du = \frac{3}{4} \int \frac{1}{u} du \]This is a natural logarithmic function:\[ = \frac{3}{4} \ln|u| + C \]

Step 5: Substitute Back in Terms of x

Now substitute back \(u = 1 + 2x^2\) to express the result in terms of \(x\):\[ \frac{3}{4} \ln|u| + C = \frac{3}{4} \ln|1 + 2x^2| + C \]

Step 6: Finalize and Simplify

The integral simplifies nicely in its final form, taking into account the constant of integration \(C\). Thus, the evaluated indefinite integral is:\[ \frac{3}{4} \ln|1 + 2x^2| + C \]

Key Concepts

Indefinite Integral

An indefinite integral is a fundamental concept in calculus representing a family of functions. It's often described as the reverse process of taking a derivative. The symbol used is the integral sign: \( \int \), followed by an expression and \( dx \), which indicates integration with respect to \( x \). Indefinite integrals do not have upper and lower limits, leading to a result that includes a constant of integration, symbolized by \( C \). This constant accounts for the fact that multiple functions can have the same derivative.

The indefinite integral provides a general solution to a differential equation. Each member of this family of functions differs by a constant value. For example, the indefinite integral of \( f(x) = x \) would be \( \int x \, dx = \frac{x^2}{2} + C \).

• The integral sign indicates the operation of integration.
• \( dx \) specifies the variable being integrated.
• The constant \( C \) is crucial as it allows for all possible antiderivatives of the function.

Substitution Method

The substitution method is a technique used to simplify the process of finding an integral. It involves replacing a particular expression in the integral with a new variable. This is particularly helpful when the integral contains a function and its derivative. The goal is to make the integral simpler, turning it into a new one that is easier to solve.

In the given exercise, the substitution involved setting \( u = 1 + 2x^2 \), which simplifies the process by aligning the denominator's derivative with part of the numerator. Once we differentiated \( u \), it was replaced into the original integral, allowing for straightforward integration.

• Choose a substitution that simplifies one part of the integral.
• Align the derivative of the substitution with another part of the integral.
• Transform and integrate the simplified expression.
• Convert back to the original variable once integrated.

Natural Logarithm

The natural logarithm is a logarithmic function denoted as \( \ln \). It's the inverse of the exponential function \( e^x \), where \( e \) is approximately equal to 2.718. In calculus, it's significant when integrating functions of the form \( \frac{1}{u} \).

In the exercise above, the function inside the integral was transformed into \( \frac{1}{u} \) through substitution, leading to the integral \( \int \frac{1}{u} \, du \), which naturally evaluates to \( \ln|u| + C \). The absolute value is necessary because a logarithm of a negative number is undefined.

• The natural logarithm simplifies integrals involving \( \frac{1}{x} \).
• It often results in solutions expressed in terms of \( \ln \).
• Remember the absolute value to ensure correctness across all x values.

Integral Evaluation

Integral evaluation refers to the process of calculating the definite or indefinite integral of a function. For indefinite integrals, this means finding the antiderivative and including a constant of integration, \( C \). In definite integrals, we solve for a numerical value between specified limits.

In our exercise, after setting up the integral with substitution, integral evaluation involved integrating the simplified form \( \int \frac{3}{4u} \, du \). This was followed by replacing \( u \) back with \( 1 + 2x^2 \).

The steps in evaluating an indefinite integral generally include:

• Performing substitutions to simplify complexity.
• Converting the expression into a manageable form, like a standard integral.
• Integrating each part step-by-step.
• Substituting back to the original variable to finalize the expression.

In the end, we add the constant \( C \) as this represents the family of all possible solutions.