Problem 91 - Thomas Calculus in SI Units

Question

Evaluate the integrals in Exercises \(83-92.\) $$\int_{2}^{4} x^{2 x}(1+\ln x) d x$$

Step-by-Step Solution

Answer

The integral evaluates to 32760.

Step 1: Identify the Integral Type

The given integral is \( \int_{2}^{4} x^{2x}(1 + \ln x) \, dx \), which is a definite integral with a complex exponential function \( x^{2x} \) multiplied by \( 1 + \ln x \). This type of integral is best approached using substitution, recognizing patterns in mathematical functions and logarithms.

Step 2: Choose an Appropriate Substitution

Let \( u = x^{2x} \). To find \( du \), differentiate: \( u = e^{2x \cdot \ln x} \) so that \( \ln u = 2x \ln x \). Differentiating, \( \frac{du}{dx} = e^{2x \ln x} (2 \ln x + 2) = 2x^{2x} (\ln x + 1) \).

Step 3: Rewrite the Integral with the Substitution

Using the substitution, the differential \( du = 2x^{2x}(\ln x + 1) \, dx \) simplifies the integral to \( 0.5 \int du \) over the limits transformed by \( u(2) \) and \( u(4) \).

Step 4: Evaluate the Substituted Integral

This substitution results in: \[ \int_{2}^{4} x^{2x} (1 + \ln x) \, dx = 0.5 \int du \] over its corresponding transformed limits. Evaluate \( 0.5(u(4) - u(2)) \).

Step 5: Substitute Back and Calculate Limits

Evaluate \( u(x) = x^{2x} \, \text{at} \, x = 4 \, \text{and} \, x = 2 \): \( u(4) = 4^{8} \) and \( u(2) = 2^{4} \). Hence, integral equals \( \frac{1}{2}(4^8 - 2^4) \). Calculate the numerical result.

Step 6: Perform Final Calculations to Get Result

Calculate \( 4^8 = 65536 \) and \( 2^4 = 16 \), resulting in the integral: \( \frac{1}{2}(65536 - 16) = \frac{1}{2}(65520) = 32760 \). Hence, the final answer is 32760.

Key Concepts

Definite Integral

A definite integral is used to calculate the area under a curve between two specific points. In the given exercise, we have a definite integral from 2 to 4, represented by the integral limits. When we talk about definite integrals, they are accompanied by two main components: the integrand and the limits of integration.

• The integrand is the function we aim to integrate, which in this case is the product of two functions: an exponential and a logarithmic function.
• The limits, here 2 and 4, define where on the x-axis we measure the area under the curve of the integrand.

The process of calculating this area involves several steps but is often simplified by using techniques such as substitution when the function is complex. The definite integral results in a numerical value that represents the area, which is 32760 for the given problem.

Integral Substitution

Integral substitution, sometimes referred to as the substitution method, is a common technique for simplifying integrals, by changing variables. This approach is particularly useful when the integral involves complex functions, which are hard to integrate directly.
For this exercise, the integrand involves the function \(x^{2x}(1 + \ln x)\), which can be daunting. We apply substitution by letting \( u = x^{2x} \). This changes the original variable \( x \) to a new variable \( u \), simplifying the integral.

• First, express \( u \) in terms of \( x \). For \( u = x^{2x} \), find \( du \) by differentiating \( u \) with respect to \( x \).
• Rewrite the original integral in terms of \( u \), which often makes it much more approachable.

Substitution shifts the complex function to a simpler form, in this case leading to \( 0.5 \int du \), and makes the process of integration more straightforward.

Exponential Function

Exponential functions grow extremely fast, making them intriguing yet sometimes challenging to integrate. An exponential function is any function that has the variable in the exponent. Our original problem's integrand features an exponential function \( x^{2x} \), a composite function including both exponential and polynomial characteristics.
Exponential functions include elements such as:

• Rapid growth, meaning the value increases exponentially with an increase in the input variable.
• Appear frequently in real-world scenarios, from population growth to radioactive decay.

Integrating such functions often requires substitutions, and understanding the derivative of the function helps in determining the substitution matrix, like how \( u = x^{2x} \) leads to straightforward integration in our exercise.

Logarithmic Integration

Logarithmic integration deals with handling integrals containing logarithmic expressions. In our exercise, the term \( 1 + \ln x \) in the integrand can be factored as part of the substitution process. Logarithms have specific properties that simplify integration:

• The logarithm of a product can be rewritten as a sum: \( \ln(ab) = \ln a + \ln b \)
• When differentiating \(\ln x\), we get \( \frac{1}{x} \), which sometimes helps in substitution.

These properties make rewriting and simplifying expressions with logarithms manageable. When doubling up by using substitutions, like in the conversion from \( x \) to \( u \) based on logarithmic terms, the integral structure shifts into simpler units, facilitating easier computation. After proper substitution, evaluating the integral becomes more direct, confirming these techniques as essential for integrals involving logarithmic functions.