Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int \ln \left(7 x^{5}\right) d x $$

Step-by-Step Solution

Answer

The integral is \( x \ln(7x^5) - 5x + C \).

Step 1: Identify parts for integration by parts

Integration by parts formula is \( \int u \, dv = uv - \int v \, du \). We need to identify \( u \) and \( dv \) for the given integral. Choose \( u = \ln(7x^5) \) and \( dv = dx \).

Step 2: Differentiate and integrate to find \( du \) and \( v \)

Differentiate \( u \) to find \( du \): \( du = \frac{d}{dx}(\ln(7x^5)) dx = \frac{1}{7x^5} \cdot 35x^4 \, dx = \frac{5}{x} \, dx \). Integrate \( dv \) to find \( v \): \( v = \int dx = x \).

Step 3: Substitute into integration by parts formula

Using the integration by parts formula: \( \int \ln(7x^5) dx = uv - \int v \, du \). Substitute \( u \), \( du \), \( v \), and \( dv \) found earlier: \( = x \ln(7x^5) - \int x \cdot \frac{5}{x} dx \).

Step 4: Simplify the integral

Simplify the integral part: \( \int x \cdot \frac{5}{x} dx = \int 5 \, dx \). This simplifies to \( 5x \).

Step 5: Combine results and write final answer

Now substitute back the simplified integral: \( x \ln(7x^5) - 5x + C \), where \( C \) is the constant of integration.

Key Concepts

Natural Logarithms

Natural logarithms are a fundamental concept in mathematics, widely used in calculus and higher-level math. They are the logarithms that have the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. In practical terms, natural logarithms are used to solve exponential growth problems and occur naturally in many areas like finance, physics, and biology.

When dealing with integrals involving natural logarithms, it's crucial to understand how the derivative of a natural logarithm works. The derivative of \( \ln(x) \) is \( \frac{1}{x} \), which is an essential fact used in differentiation and integration involving natural logarithmic functions.

• The natural logarithm function, \( \ln(x) \), is defined only for positive real numbers.
• It has a vertical asymptote at \( x = 0 \).
• It increases monotonically, meaning it's always increasing as the input value grows.

This understanding helps in applying integration techniques like integration by parts, especially when natural logarithms are part of the function to integrate.

Differentiation

Differentiation is a core concept in calculus that deals with finding the derivative of a function. The derivative measures how a function's output value changes as its input changes. In the context of the original exercise, differentiation is used to find the derivative of \( u = \ln(7x^5) \) in the integration by parts method.

To differentiate \( \ln(7x^5) \), apply the chain rule, which is a method for finding the derivative of a composition of functions. The derivative \( \frac{d}{dx} \ln(7x^5) \) involves differentiating the outer function, the natural logarithm, followed by the inner function, \( 7x^5 \).

• First, differentiate the natural logarithm: \( \frac{d}{dx} \ln(u) = \frac{1}{u} \).
• Apply it to the inner part: \( \frac{1}{7x^5} \).
• Next, differentiate the inside: \( \frac{d}{dx}(7x^5) = 35x^4 \).
• Multiply them together: resulting in \( \frac{5}{x} \).

Differentiation is a crucial tool not only for solving the problem but also for understanding the behavior of changing quantities in calculus.

Definite Integrals

Definite integrals are used to find the area under a curve between two points. It's a concept fundamental to integral calculus, representing cumulative sums where the limits of integration indicate the interval. In the context of integration by parts, even though the original exercise focuses on indefinite integrals, understanding definite integrals enriches comprehension of the broader applicability of integration techniques.

The definite integral is typically represented as \( \int_a^b f(x) \, dx \), demonstrating the integral of \( f(x) \) from \( a \) to \( b \). This results in a specific number, unlike an indefinite integral, which results in a family of functions.

• Definite integrals add precision by calculating exact values over a specific interval.
• They are used widely in real-world applications such as calculating distances, areas, and volumes.
• The Fundamental Theorem of Calculus connects differentiation and integration, establishing that the inverse of differentiation is integration.

Understanding definite integrals enhances your grasp of how integration processes translate to tangible quantities and their significance in practical scenarios.