Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Use the Log Rule to find the indefinite integral. $$ \int \frac{x}{x^{2}+1} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\frac{x}{x^2 + 1}\) dx is \(\frac{1}{2} \ln|x^2 + 1| + C\).

Step 1: Set Up Substitution

Let's make a substitution. We'll set \(u\) equal to \(x^2 + 1\). Write this as \(u = x^2 + 1\).

Step 2: Differentiate the Substitution and Solve for dx

Next, differentiate \(u\) with respect to \(x\) to get \(\frac{du}{dx} = 2x\). We then solve for \(dx\), yielding \(dx = \frac{du}{2x}\).

Step 3: Substitute Back into the Integral

We then substitute into our integral, replacing \(x^2 + 1\) with \(u\), and \(dx\) with \(\frac{du}{2x}\). This results in \(\int \frac{x}{u} *\frac{du}{2x}\).

Step 4: Simplify the Integral

The \(x\) in the numerator and the \(x\) in the denominator cancel out, leaving \(\frac{1}{2} \int \frac{du}{u}\).

Step 5: Compute the Integral

The integral of \(\frac{1}{u}\) with respect to \(u\) is \(\ln|u|\), so computing the integral gives \(\frac{1}{2} \ln|u| + C\).

Step 6: Substitute Back Original Variables

Substitute back the original variable in place of \(u\). This results in \(\frac{1}{2} \ln|x^2 + 1| + C\).

Key Concepts

Log Rule

The Log Rule is an essential concept in calculus, allowing us to integrate functions of the form \(\frac{1}{u}\). Here, \(u\) must be a function of \(x\). This rule states that if you have an integral like \(\int \frac{1}{u} \,du\), the result is \(\ln|u| + C\), where \(C\) is the constant of integration. The absolute value is crucial because the natural logarithm is only defined for positive values. Thus,

• For any integrand of the form \(\frac{1}{u}\), use the Log Rule.
• Ensure \(u\) is differentiated correctly.
• Don't forget the constant \(C\).

In the problem, after substitution and simplification, the integral becomes \(\frac{1}{2} \int \frac{1}{u} \,du\). Applying the Log Rule, it evaluates to \(\frac{1}{2} \ln|u| + C\). This simple yet powerful rule is beneficial when dealing with rational functions like \(\frac{x}{x^2+1}\).

Substitution Method

The substitution method is a clever technique used to simplify integrals by changing variables. It involves substituting part of the integrand with a single new variable. This step often simplifies the integral, making it easier to solve. Here’s how you generally apply substitution:

• Choose a substitution. Look for a part of the integrand that, when simplified, becomes easy to integrate.
• Differentiate your substitution, usually expressed as \(u\), and solve for \(dx\).
• Replace all occurrences of the original variable with the new variable in your integral.

For the given exercise, by setting \(u = x^2 + 1\), the integral transforms substantially. Differentiating gives \(du = 2x dx\), or \(dx = \frac{du}{2x}\). This substitution eliminates the variable \(x\) from the integral, allowing focus on \(u\). This method is particularly useful when dealing with complex expressions where a direct integration is not straightforward.

Integration Techniques

Integration techniques are various methods used to solve integrals, tailored to different forms and complexities of the expressions. These can include:

• Basic Integration: Recognizing standard forms and directly applying basic rules.
• Substitution: Changing variables to simplify the integral, as demonstrated in this exercise.
• Integration by Parts: Useful for products of functions, using the formula \(\int u \,dv = uv - \int v \,du\).
• Partial Fractions: Breaking down complex fractions into simpler parts for easier integration.
• Trigonometric Integrals: When integrals involve trigonometric functions, applying identities can simplify the process.

In this exercise, substitution was the key technique, as it rendered the integrand into a simpler form involving the Log Rule. A clear understanding and strategic choice of techniques can make solving integrals more manageable and greatly simplify the solving process.