Problem 43 - Applied Calculus

Question

Find \(\int 4 x\left(x^{2}+1\right) d x\) using two methods: (a) Do the multiplication first, and then antidifferentiate. (b) Use the substitution \(w=x^{2}+1\). (c) Explain how the expressions from parts (a) and (b) are different. Are they both correct?

Step-by-Step Solution

Answer

Both methods are correct; they differ by a constant.

Step 1: Multiplication – Method (a)

First, expand the expression given in the integral. Multiply \(4x\) by \(x^2 + 1\): \[ 4x(x^2 + 1) = 4x^3 + 4x. \] Now, rewrite the integral as: \( \int (4x^3 + 4x) \, dx \).

Step 2: Integration – Method (a)

Integrate the expression term by term: \[ \int 4x^3 \, dx = x^4 + C_1 \] and \[ \int 4x \, dx = 2x^2 + C_2. \] Combine the results with a single constant \( C \) to get: \[ x^4 + 2x^2 + C. \]

Step 3: Substitution Setup – Method (b)

For substitution, let \( w = x^2 + 1 \). Then, differentiate \( w \, \) to find \( dw \): \[ dw = 2x \, dx \] or \( x \, dx = \frac{1}{2} \, dw \). Substitute back into the integral: \[ \int 4x(x^2 + 1) \, dx = 2 \int w \, dw. \]

Step 4: Integration – Method (b)

Now integrate the expression with respect to \( w \): \[ 2 \int w \, dw = 2 \left( \frac{w^2}{2} \right) + C = w^2 + C. \] Substitute back \( w = x^2 + 1 \) to get the result: \[ (x^2 + 1)^2 + C. \]

Step 5: Comparison and Correctness Check

Both results, even though they appear different, are correct. Expanding \( (x^2 + 1)^2 \), you get \( x^4 + 2x^2 + 1 \). Thus, both methods lead to expressions that only differ by a constant, which is acceptable in indefinite integrals.

Key Concepts

Indefinite Integrals

Indefinite integrals represent the inverse process of differentiation. They are known in mathematics as antiderivatives, and the integral of a function involves finding a function whose derivative is the original function. When solving for indefinite integrals, we always include a constant of integration, typically represented by 'C'.
The reason we add this constant is because the derivative of a constant is zero, and it accounts for any vertical shifts in the graph of the antiderivative.

• The expression for an indefinite integral is written as \( \int f(x) \, dx \), where \( f(x) \) is the function being integrated.
• The result of an indefinite integral is a function or a family of functions, together with the constant \( C \).

In the exercise, the indefinite integrals were obtained using different methods but the results are the same except for a constant. This demonstrates the flexibility and power of integral calculus in finding solutions that differ only by a constant.

Substitution Method

The substitution method is a technique used to simplify the process of finding the integral of a more complex function. This method involves changing the variable of integration to a new variable that makes the integral easier to solve. By performing substitution, integrals can often be transformed into a simpler form.
In the context of the exercise, we let \( w = x^2 + 1 \), which simplified the function to integrate. We then compute the differential \( dw \) in terms of \( dx \) and substitute into the original integral. This substitution transformed the integral into a simpler form, allowing us to integrate with respect to \( w \) easily.

• Choose a substitution that simplifies the integral.
• Replace the original variable and differential with the new variable and its differential.
• Integrate the new function.
• Convert back to the original variable to obtain the final answer.

In this exercise, substitution proved to be a powerful tool for simplifying and solving the integral.

Polynomial Integration

Polynomial integration involves finding the integral of polynomial functions. A polynomial is a sum of terms, each consisting of a variable raised to a non-negative integer power, multiplied by a coefficient.
This type of integration is often straightforward as each term of the polynomial can be integrated separately.
When integrating a polynomial, the basic rule to follow is:

• For a term \( ax^n \), the integral is \( \frac{a}{n+1}x^{n+1} + C \), provided \( n eq -1 \).

In the exercise, polynomial integration was used after multiplying and expanding the expression \( 4x(x^2 + 1) \) to obtain \( 4x^3 + 4x \). Each term was integrated separately, which resulted in \( x^4 \) and \( 2x^2 \), combining them gives the polynomial integral with the constant \( C \). Polynomials can be integrated term by term, making them simpler when using this approach.