Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. $$\int_{1}^{2} \frac{4}{9 x^{2}+6 x+1} d x$$

Short answer

Question: Evaluate the definite integral $$\int_{1}^{2} \frac{4}{9 x^{2}+6 x+1} d x$$. Answer: The value of the definite integral is $$\frac{1}{7}$$.

Step-by-step solution

Identify the substitution

We need to find a suitable substitution to simplify the integral. Notice the denominator \(9x^2 + 6x + 1\). We can represent this as \((3x + 1)^2\). So, we will make the substitution \(u = 3x + 1\).

Differentiate the substitution and find dx

Now, differentiate the substitution with respect to x to find the differential: $$\frac{d u}{d x} = 3 \implies d u = 3 d x$$ Thus, \(dx = \frac{du}{3}\).

Change the limits of integration

Before moving forward, let's change the limits of integration to match our substitution. When \(x = 1\), \(u = 3(1) + 1 = 4\). When \(x = 2\), \(u = 3(2) + 1 = 7\). So, our new limits of integration are \(u = 4\) and \(u = 7\).

Perform the substitution

Now, replace the original integral with the new variables and limits: $$\int_{4}^{7} \frac{4}{u^2} \cdot \frac{du}{3}$$ Simplify the integrand: $$\int_{4}^{7} \frac{4}{3} u^{-2} d u$$

Evaluate the integral

Evaluate the antiderivative of the simplified integrand: $$\frac{4}{3} \int_{4}^{7} u^{-2} d u = \frac{4}{3} \left[ -u^{-1} \right]_4^7$$

Apply the limits of integration

Now, apply the limits of integration to the antiderivative: $$\frac{4}{3} \left[-\frac{1}{7} - (-\frac{1}{4}) \right] = \frac{4}{3} \left[\frac{1}{4} - \frac{1}{7} \right] = \frac{4}{3} \cdot \frac{3}{28} = \frac{12}{84}$$

Simplify the result

Simplify the result: $$\frac{12}{84} = \frac{1}{7}$$ Therefore, the value of the definite integral is: $$\int_{1}^{2} \frac{4}{9 x^{2}+6 x+1} d x = \frac{1}{7}$$

Key concepts

Integration by Substitution

Integration by substitution is a powerful technique that simplifies complex integrals. Think of it as a method for changing the variable to make integration more manageable. In this exercise, we encountered the expression \(9x^2 + 6x + 1\). To make it simpler, we choose a substitution: let \(u = 3x + 1\). This reduces the complexity by transforming the integral into a simpler form.
Here are the basic steps for using substitution:

• Select the substitution: Choose \(u = f(x)\) to simplify the integrand.
• Differentiate to find \(dx\): Solve for \(dx\) in terms of \(du\) and simplify.
• Rewrite the integral: Substitute \(u\) and \(dx\) into the integral.
• Integrate and back-substitute: Perform the integration in \(u\) and revert to \(x\) if needed.

Mastering substitution can turn challenging integrals into easy ones. Just remember to replace every \(x\) variable in the integral.

Change of Variables

Changing variables is an essential part of integration by substitution. When we switch from \(x\) to \(u\), we aren’t just swapping letters. We’re adjusting the whole setup of our integral to make the math easier. In our example, the transformation was from \(x\) to \(u = 3x + 1\).
Here is why changing variables is so helpful:

• It aligns the integral with familiar or simpler forms, like basic power rules.
• Keeps the algebra manageable by reducing complex expressions.
• Adapts the limits of integration to fit the substitution.

By carefully converting both the function and the limits, the integration becomes straightforward. The change in variables also ensures consistency and correctness in calculations.

Limits of Integration

After choosing a substitution, it’s crucial to update the limits of integration. The limits define the range where the integral is evaluated. In our example, \(x = 1\) becomes \(u = 4\) and \(x = 2\) becomes \(u = 7\). This ensures that the integral's range fits the new variable.
Here's how to change the limits:

• Substitute the original limits into the substitution equation.
• Calculate the new limits based on these results.
• Always double-check that the transformed limits correspond to the correct range with \(u\).

Correct limits prevent errors and ensure accurate results when calculating definite integrals. This step is essential for applying the Fundamental Theorem of Calculus correctly.

Antiderivative Calculation

Finding the antiderivative is a crucial step in evaluating integrals. After substitution and changing limits, focus on the antiderivative of the simplified function. In our example: \[ \int u^{-2} \, du \] The antiderivative here is straightforward using the power rule: \[ -\frac{1}{u} \] Remember these key points about antiderivatives:

• Identify the form of the function to use the correct rule.
• For powers of \(u\), use the power rule: \( \int u^n \, du = \frac{u^{n+1}}{n+1} \) (except \(n = -1\)).
• Evaluate the antiderivative at the new limits and subtract.

Calculating the antiderivative accurately leads to finding the definite integral's value. This wraps up the problem, giving us the final result \(\frac{1}{7}\).