Problem 57 - Thomas Calculus in SI Units

Question

Verify the formulas by differentiation. $$\int(7 x-2)^{3} d x=\frac{(7 x-2)^{4}}{28}+C$$

Step-by-Step Solution

Answer

The integration formula is correct as differentiating the result yields the original integrand.

Step 1: Understand the Goal

We need to verify the integration result by differentiating the right-hand side of the given equation. This means taking the derivative of \( \frac{(7x-2)^4}{28} + C \) and checking if it equals the integrand \((7x-2)^3\).

Step 2: Differentiate the Constant Term

Since differentiation of a constant \(C\) yields zero, we don't have to consider it when differentiating \( \frac{(7x-2)^4}{28} + C \). Thus, focus solely on differentiating \( \frac{(7x-2)^4}{28} \).

Step 3: Apply the Chain Rule

To differentiate \( \frac{(7x-2)^4}{28} \), apply the chain rule. Let \( u = 7x - 2 \), then\( \frac{d}{dx}(u^4) = 4u^3 \cdot \frac{du}{dx} \). Here, \( \frac{du}{dx} = 7 \).

Step 4: Simplify the Derivative Expression

Substitute back \( u = 7x - 2 \) to get the derivative: \[ \frac{d}{dx} \left( \frac{(7x-2)^4}{28} \right) = \frac{1}{28} \cdot 4(7x-2)^3 \cdot 7 \] Simplify this to \( \frac{28(7x-2)^3}{28} = (7x-2)^3 \).

Step 5: Verify Final Result

Since the differentiated result \((7x-2)^3\) matches the original integrand, the given integral formula is verified.

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to find the derivative of composite functions. When you have a function inside another function, you use the chain rule to differentiate them. This is especially helpful when dealing with powers or complex expressions.

A simple way to remember the chain rule is: If you have a composite function like \( h(x) = f(g(x)) \), the derivative \( h'(x) \) is obtained by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function: \[ h'(x) = f'(g(x)) imes g'(x) \]

• Let the inner function be \( u \), so \( u = g(x) \).
• Find \( \frac{du}{dx} \). This is the derivative of the inside function \( g(x) \).
• Find \( \frac{d}{du}(f(u)) \). This is the derivative of the outside function \( f(g(x)) \).
• Multiply these derivatives together.

When applied correctly, the chain rule simplifies the differentiation of complex expressions.

Integration

Integration is the process of finding the integral of a function, essentially the reverse operation of differentiation. It is often used to determine the accumulation of quantities, such as areas under curves. The integration of a function \( f(x) \) with respect to \( x \) can be represented as \( \int f(x) \, dx \).

There are various integration techniques, including:

• Basic antiderivatives
• Substitution
• Integration by parts
• Partial fraction decomposition

In the example provided, we use basic antiderivatives to integrate the expression \((7x-2)^3\). By integrating, we find \( \frac{(7x-2)^{4}}{28} + C \). The constant \( C \) is added because integration can have many different functions that differ by a constant.

Derivative

A derivative represents the rate at which a function is changing at any given point. It provides the slope of the function’s curve at a particular point. In mathematical terms, if you have a function \( y = f(x) \), the derivative \( f'(x) \) is calculated as the limit of the average rate of change of \( f \) as the interval approaches zero.

In the process of differentiation, certain rules apply, such as:

• Power rule: \( \frac{d}{dx}(x^n) = nx^{n-1} \)
• Product rule: \( \frac{d}{dx}(uv) = u'v + uv' \)
• Quotient rule: \( \frac{d}{dx}(\frac{u}{v}) = \frac{u'v - uv'}{v^2} \)
• Chain rule: Used for composite functions.

In our example, we used the chain rule to find the derivative of \( \frac{(7x-2)^4}{28} \), applying it to the inner and outer functions to match the original integrand.

Integration Verification

Integration verification is the process of confirming the accuracy of an integral by differentiating the integrated expression. If the result of differentiation matches the original integrand, the integration is considered verified and correct.

To perform integration verification:

• Start with the integrated expression.
• Differentiate it using appropriate rules, such as the chain rule where necessary.
• Simplify the derivatives to see if they reflect the original function.

In our exercise, we verified the integral \( \int(7x-2)^3 dx \) by differentiating the result \( \frac{(7x-2)^4}{28} + C \). When simplified, it matched the original integrand, confirming the integration's validity. It's a powerful method for checking your work and ensuring accuracy in calculus.