Problem 54 - Essential Calculus

Question

Find the indefinite integral. $$ \int(3-x) 7^{(3-x)^{2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( (3-x) \cdot 7^{(3-x)^{2}} \) is \( 7^{(3-x)^{2}} + C \).

Step 1: Identify the function and its derivative

Looking at the function \(\int (3-x) \cdot 7^{(3-x)^{2}} dx \), it can be identified that the function \(f(u) = 7^{u^{2}}\) and its derivative \(f'(u) = (3-x)\) is present.

Step 2: Confirm by taking the derivative

To double check, let's compute the derivative of \(f(u)\). It's a composite function, so the chain rule is used: \(f'(u) = 2u \cdot 7^{u^{2}} \cdot ln(7)\). When \( u = 3-x \), the derivative will be \( -2x \cdot 7^{(3-x)^{2}} \cdot ln(7) \). This matches with the function to integrate, after compensating for the constant factor \(-1/(2ln(7))\).

Step 3: Apply the Fundamental Theorem

The Fundamental Theorem of Calculus states that the integral of \( f'(u)du \) over an interval \([a, b]\) is \(f(b) - f(a)\). Here, we have an identical situation, except we're finding the indefinite integral, which is a function. Therefore, the indefinite integral of \( f'(u) \) is simply \( f(u) + C \), where \( C \) is the integration constant.

Step 4: Compute the indefinite integral

Replacing \( u \) by \(3-x\), the indefinite integral becomes \( 7^{(3-x)^{2}} + C \)

Key Concepts

Integration by Substitution

One of the quintessential techniques in calculus is integration by substitution, often deemed the reverse process of the chain rule in differentiation. This method is particularly beneficial when dealing with integrals of composite functions, where a section of the integrand and its derivative are noticeable in the integral.

For the exercise \( \int(3-x) 7^{(3-x)^{2}} dx \), we identify that the integrand consists of two parts: the function \( f(u) = 7^{u^{2}} \) and its corresponding derivative \( f'(u) = (3-x) \). The substitution \( u = 3 - x \) simplifies the integral since \( du = -dx \) mirrors \( f'(u) \) (after adjusting for negative signs and constants).

Thus, the process goes as follows: Identify \( u \) and \( \frac{du}{dx} \) (derivative of \( u \) with respect to \( x \)), make the substitution, solve the integral in terms of \( u \) and, finally, replace \( u \) back with \( 3 - x \) to complete the integration by substitution.

The Chain Rule

A pivotal tool in calculus is the chain rule, used when differentiating composite functions. In essence, it allows for the differentiation of a function of a function. The chain rule states that if \( y = f(u) \) and \( u = g(x) \) where \( f \) and \( g \) are both differentiable, then \( y \) is differentiable and \( y' = f'(u) \cdot g'(x) \).

In the context of our exercise, the chain rule is applied in reverse as part of the integration process. Initially, confirming the derivative of \( f(u) \) with \( u = 3 - x \), we found that \( f'(u) \) involved applying the chain rule, which gave \( -2(3-x)7^{(3-x)^2} \cdot \ln(7) \) after differentiation. This confirmation is pivotal, as it indicates the presence of the function \( f(u) \) and its derivative \( f'(u) \) within the integrand, justifying the use of substitution.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a cornerstone in understanding the relationship between differentiation and integration. It bridges the concept of an antiderivative with the area under a curve. The theorem essentially states that if \( F \) is an antiderivative of \( f \) over an interval \( [a, b] \), then \( \int_a^b f(x) dx = F(b) - F(a) \).

For indefinite integrals, as seen in our exercise, the theorem implies that \( \int f'(u)du \) gives us \( F(u) + C \) where \( C \) is the integration constant, a concept crucial in solving for the indefinite integral presented in the problem. In this scenario, after confirming our function and its derivative, we apply the theorem to revert back from the derived function to the original function, inclusive of the constant \( C \).

Composite Function

A composite function, constructed by combining two functions, is a function \( f(g(x)) \), where \( g \) is applied first and then \( f \) is applied to the result. In the realm of calculus, it is fundamental to understand composite functions for both differentiation and integration purposes.

In the given integral, \( 7^{(3-x)^{2}} \) represents a composite function where \( g(x) = 3-x \) and \( f(u) = 7^{u^2} \). Recognizing this structure is vital to apply appropriate techniques like the chain rule for differentiation or substitution for integration. It's through comprehending the nature of composite functions that we can break down more complex expressions into simpler parts, enabling us to solve them with greater ease.