Problem 1 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int\left(8+x^{3}\right)^{5} 3 x^{2} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{(8 + x^3)^6}{6} + C \).

Step 1: Identify the Inner Function

The inner function can be identified as \( u = 8 + x^3 \), which will be crucial for applying the substitution method.

Step 2: Differentiate the Inner Function

Differentiate \( u = 8 + x^3 \) with respect to \( x \), yielding \( \frac{du}{dx} = 3x^2 \). This suggests that \( du = 3x^2 \, dx \).

Step 3: Substitute in Terms of 'u'

Rewrite the integral using the substitution \( u = 8 + x^3 \). This gives \( \int u^5 \, du \), because \( 3x^2 \, dx = du \).

Step 4: Integrate with Respect to 'u'

Now integrate \( \int u^5 \, du \). The antiderivative of \( u^5 \) is \( \frac{u^6}{6} + C \), where \( C \) is the constant of integration.

Step 5: Substitute Back for 'u'

Substitute \( u = 8 + x^3 \) back into the expression to get \( \frac{(8 + x^3)^6}{6} + C \).

Step 6: Differentiate to Check

Differentiate the solution \( \frac{(8 + x^3)^6}{6} + C \) with respect to \( x \). Applying the chain rule, we get \( 6 \cdot (8 + x^3)^5 \cdot x^2 \) divided by 6, which simplifies to \( (8 + x^3)^5 \, 3x^2 \), matching the integrand.

Key Concepts

Substitution Method

The substitution method is a powerful technique used to simplify complex integrals, by transforming them into an easier form. This is especially handy when dealing with composite functions. The main idea is to choose a substitution that makes the integral simpler to evaluate. Here’s how it works:

• Identify the inner function within the integrand, which you’ll replace with a new variable. In our problem, we started with the integral \(\int (8+x^3)^5 3x^2 \, dx\), and the inner function was \(u = 8 + x^3\).
• Differentiate the inner function to find \(du\) in terms of \(dx\). Thus, \(\frac{du}{dx} = 3x^2 \), which gives \(du = 3x^2 \, dx\).
• Substitute \(u\) into the integral and express everything in terms of \(u\) and \(du\). This results in an easier integral: \(\int u^5 \, du\).

By using substitution, you transform an integral that is difficult to solve directly into one that is straightforward, making it a valuable tool for calculus students.

Antiderivative

An antiderivative is the reverse process of differentiation, aiming to determine the original function from its derivative. When you find an antiderivative, you're essentially solving for the function that results in the given derivative.

• In the substitution method, once the integral is rewritten, we need to find the antiderivative of the simpler function. For \(\int u^5 \, du\), the antiderivative is \(\frac{u^6}{6} + C\), where \(C\) is the constant of integration.
• This fundamental concept helps transform the act of integration into a simpler form of finding the antiderivative of basic functions.

Antiderivatives are crucial since they allow us to reconstruct the original function, providing a connection between differentiation and integration. Every definite integral process aims to find a function's antiderivative before evaluating it.

Chain Rule

The chain rule is an essential calculus tool for differentiating composite functions. It is especially useful when verifying solutions, like in our case, where it ensures our integral evaluation was correct.

• To check our integral solution, we differentiated \(\frac{(8 + x^3)^6}{6} + C\). Using the chain rule, \(\frac{d}{dx}\) of a composite function \(f(g(x))\) becomes \(f'(g(x)) \cdot g'(x)\).
• In our example, \((8+x^3)^6\) was differentiated to \(6(8+x^3)^5 \cdot 3x^2 \), which eventually simplified back to \((8+x^3)^5 \cdot 3x^2\), the original integrand.

The chain rule is indispensable for confirming that we have indeed found the correct antiderivative. It links our integration work back to differentiation, closing the loop on the problem-solving process.