Problem 22 - Calculus Early Transcendentals

Question

Use a change of variables to find the following indefinite integrals. Check your work by differentiating. $$\int \frac{1}{10 x-3} d x$$

Step-by-Step Solution

Answer

#Short Answer# The integral of the given function is: $$\int\frac{1}{10x-3}dx=\frac{1}{10}\ln{|10x-3|}+C$$

Step 1: Introduce a new variable

Let us introduce a new variable \(u\) by using substitution: $$u = 10x - 3$$ Now, we need to find the corresponding \(d u\) in terms of \(d x\); to do this, differentiate both sides with respect to \(x\): $$\frac{d u}{d x} = 10$$ Therefore, $$d u = 10 d x$$ Now, we are ready to perform the substitution for both \(u\) and \(d u\).

Step 2: Substitute and integrate

Using substitution, we rewrite the integral as: $$\int\frac{1}{10x-3}dx=\int\frac{1}{u} \frac{d u}{10}$$ This can be further simplified to: $$\frac{1}{10}\int\frac{1}{u}d u$$ Now, we can integrate the expression: $$\frac{1}{10}\int\frac{1}{u}d u = \frac{1}{10}\ln{|u|}+C$$, where C is the constant of integration.

Step 3: Substitute back in terms of x

Since we initially made a substitution, we now need to revert back to the original variable \(x\). So, replace u back: $$\frac{1}{10}\ln{|10x - 3|} + C$$ So, the integral is: $$\int\frac{1}{10x-3}dx=\frac{1}{10}\ln{|10x-3|}+C$$

Step 4: Check the result by differentiating

To check our work, we will differentiate the result and ensure that it matches the integrand. Using the chain rule, differentiate the obtained result with respect to x: $$\frac{d}{d x}\left(\frac{1}{10}\ln{|10x-3|}\right) = \frac{1}{10} \cdot \frac{1}{10x-3} \cdot 10$$ which simplifies to: $$\frac{1}{10x-3}$$ Our differentiation matches the given integrand, which confirms that our solution is correct.

Key Concepts

Change of Variables

The change of variables technique is a strategic way to simplify complex integrals. It involves introducing a new variable that can make the integration more manageable. In this context, we refer to this as substitution.

When we introduce a new variable, typically labeled as \(u\), we express some part of the original integrand in terms of this new variable.

• The purpose is to transform the integral into a simpler form that is easier to solve.
• It often requires a corresponding differential transformation, where \(d x\) is expressed as \(d u\) based on the derivative of \(u\) with respect to \(x\).

For example, in the given exercise, we set \(u = 10x - 3\), which alters the problem into something more straightforward to integrate.

This method is particularly useful when dealing with composite functions or where the inner function is reasonably isolated.

Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus, representing the family of all antiderivatives of a function. Unlike definite integrals, which calculate the area under a curve between two points, indefinite integrals do not have limits of integration.

Mathematically, an indefinite integral is represented as:\[\int f(x) \, dx\]This notation indicates the process of finding all functions whose derivative is \(f(x)\).

The solution includes a constant, \(C\), known as the constant of integration.

• This constant represents the infinite number of vertical shifts possible for antiderivatives, as differentiation loses the constant term.
• In our exercise, the result is \(\frac{1}{10}\ln{|10x - 3|} + C\) to emphasize that there are many functions whose derivative is the integrand \(\frac{1}{10x-3}\).

Chain Rule

The chain rule is a crucial differentiation technique used when dealing with composite functions. It is particularly important in validating our integration results by differentiation, as in the given problem.

The chain rule formula is:\[\frac{d}{dx}[g(f(x))] = g'(f(x)) \cdot f'(x)\]This means that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

In our integration problem, when we differentiate:\[\frac{d}{d x}\left(\frac{1}{10}\ln{|10x-3|}\right)\]

• The outer function is the natural logarithm, and its derivative is \(\frac{1}{u}\).
• The inner function is \(10x-3\), and its derivative is \(10\).
• Applying the chain rule, we find the derivative of the entire expression accords with the initial integrand \(\frac{1}{10x-3}\), confirming our solution.

Using the chain rule reinforces both the correctness and the utility of substitution in solving indefinite integrals.