Problem 65 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int_{1 / 2}^{(\sqrt{2}+3) /(2 \sqrt{2})} \frac{d x}{8 x^{2}-8 x+11}$$

Step-by-Step Solution

Answer

Answer: $\ln\left|\frac{19}{21}\right|$

Step 1: Identify a suitable substitution

Let's consider the substitution, \(u = 4x^2 - 4x + 5.5\), which would lead us to a simpler integrand. To make this work, first differentiate u with respect to x.

Step 2: Differentiate the substitution equation

Differentiate \(u\) with respect to \(x\). It can be done as follows: $$\frac{d u}{d x} = \frac{d}{d x}(4 x^{2} - 4 x + 5.5) = 8 x - 4$$ Now we need to find \(dx\) in terms of \(du\).

Step 3: Find dx in terms of du

Solve the equation above for \(dx\). This gives $$dx = \frac{du}{8x - 4}$$

Step 4: Rewrite the integral in terms of u

Now, we can rewrite the integral in terms of \(u\) using the substitutions we found: $$\int_{1 / 2}^{(\sqrt{2}+3) /(2 \sqrt{2})} \frac{d x}{8 x^{2}-8 x+11} = \int_{u(1/2)}^{u((\sqrt{2}+3) /(2 \sqrt{2}))} \frac{du}{u}$$ Notice that the bounds of integration have also been changed according to the substitution \(u\). Now, we solve for the new bounds. For the lower bound, plug in \(x = \frac{1}{2}\) into the expression for \(u\): $$u \left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^2 - 4\left(\frac{1}{2}\right) + 5.5 = \frac{9}{2}$$ For the upper bound, plug in \(x = \frac{\sqrt{2} + 3}{2\sqrt{2}}\) into the expression for \(u\): $$u \left(\frac{\sqrt{2} + 3}{2\sqrt{2}}\right) = \frac{57}{(\sqrt{2})^2+7\sqrt{2}+22} = \frac{57}{14}$$ Now, rewrite the integral with these new bounds: $$\int_{1 / 2}^{(\sqrt{2}+3) /(2 \sqrt{2})} \frac{d x}{8 x^{2}-8 x+11} = \int_{\frac{9}{2}}^{\frac{57}{14}} \frac{du}{u}$$

Step 5: Evaluate the integral in terms of u

Evaluate the integral in terms of \(u\): $$\int_{\frac{9}{2}}^{\frac{57}{14}} \frac{du}{u} = \left[\ln|u|\right]_{\frac{9}{2}}^{\frac{57}{14}} = \ln \left|\frac{57}{14}\right| - \ln \left|\frac{9}{2}\right| = \ln\left|\frac{\frac{57}{14}}{\frac{9}{2}}\right|$$

Step 6: Simplify the result

Now, simply simplify the result: $$\ln\left|\frac{\frac{57}{14}}{\frac{9}{2}}\right| = \ln \left|\frac{57}{14} \cdot \frac{2}{9}\right| = \ln \left|\frac{19}{21}\right|$$ So, the result is: $$\int_{1 / 2}^{(\sqrt{2}+3) /(2 \sqrt{2})} \frac{d x}{8 x^{2}-8 x+11} = \ln\left|\frac{19}{21}\right|$$

Key Concepts

Substitution Method

Substitution method is a powerful technique in calculus for simplifying integrals. This technique involves changing the variable of integration to simplify the integral into a more manageable form. In the exercise given, we chose a substitution, specifically

• \( u = 4x^2 - 4x + 5.5 \).

This substitution helps to transform the complex quadratic expression into a simpler term. By differentiating \( u \) with respect to \( x \), which yields \( \frac{du}{dx} = 8x - 4 \), we found a relationship between the differential \( du \) and \( dx \) as \( dx = \frac{du}{8x - 4} \).
By rearranging the integral using this substitution, we construct a new integral in terms of \( u \), which is easier to handle. The substitution method requires careful calculation of the new limits of integration, transforming the original \( x \) values into corresponding \( u \) values.

• Example calculation: Substituting the lower bound, \( x = \frac{1}{2} \), to find \( u(1/2) = \frac{9}{2} \).
• Substituting the upper bound, \( x = \frac{\sqrt{2} + 3}{2\sqrt{2}} \), results in \( u((\sqrt{2} + 3)/(2\sqrt{2})) = \frac{57}{14} \).

This conversion from \( x \) to \( u \) ensures that the substitution method is applied correctly, aiding in the simplification and evaluation of the integral.

Definite Integrals

Definite integrals calculate the net area under a curve within specific bounds. Unlike indefinite integrals, which represent a family of functions, definite integrals yield a specific numerical value.
When using definite integrals with the substitution method, it is crucial to adjust the limits of integration according to the substitution. This ensures that the area calculation corresponds correctly to the transformed variable.

• For instance, in our problem, the integration bounds transform from \( x \) values \( \frac{1}{2} \) and \( \frac{\sqrt{2} + 3}{2\sqrt{2}} \) to \( u \) values \( \frac{9}{2} \) and \( \frac{57}{14} \).

With these new bounds, the integral

• \( \int_{\frac{9}{2}}^{\frac{57}{14}} \frac{du}{u} \)

is evaluated, leading to a precise numerical solution.
When solving definite integrals, one of the most common methods of checking work is by applying the Fundamental Theorem of Calculus. This theorem indicates that to evaluate a definite integral, one can calculate the anti-derivative (indefinite integral) across the upper and lower bounds and take their difference. The proper computation provides the area under the curve and ensures all transformations align with the variable changes.

Logarithmic Integration

Logarithmic integration involves integrals resulting in a logarithmic function. This occurs frequently in expressions of the form \( \int \frac{1}{x} \, dx \). In the exercise, the integrand was transformed into \( \int \frac{du}{u} \), a classic logarithmic integral.
By evaluating this integral, the result follows the formula:

• \[ \int \frac{du}{u} = \ln|u| + C \]

Given that our context is a definite integral, the constant \( C \) becomes irrelevant. Instead, we compute the difference using the new or transposed bounds.

• Therefore, \[ \int_{\frac{9}{2}}^{\frac{57}{14}} \frac{du}{u} = \left[ \ln|u| \right]_{\frac{9}{2}}^{\frac{57}{14}} = \ln \left|\frac{57}{14}\right| - \ln \left|\frac{9}{2}\right|\].
• This simplifies as: \[ \ln \left| \frac{57}{14} \cdot \frac{2}{9} \right| = \ln \left| \frac{19}{21} \right| \].

Logarithmic integration can be particularly useful for solving integrals involving rational functions or dividing expressions where part of the solution is a simple logarithmic form. Proper application of algebraic simplifications and understanding logarithm properties are integral to solving these types of integration problems effectively.