Problem 38 - Calculus

Question

\(\int\left(x^{2}+1\right)^{2} d x\) is not \(\frac{1}{3}\left(x^{2}+1\right)^{3}\) but

Step-by-Step Solution

Answer

The integral is \( \frac{x^5}{5} + \frac{2x^3}{3} + x + C \).

Step 1: Identify the Integral Type

We aim to solve \( \int (x^2 + 1)^2 \, dx \). This integral appears to be a polynomial under a power. The naive substitution suggested is incorrect because it does not account for the chain rule influence on the derivative.

Step 2: Expand the Polynomial

Before substituting, let's expand \((x^2 + 1)^2\) using the identity \((a + b)^2 = a^2 + 2ab + b^2\). Expanding, we get: \((x^2 + 1)^2 = (x^2)^2 + 2\cdot(x^2)\cdot(1) + 1^2 = x^4 + 2x^2 + 1\).

Step 3: Integrate Each Term Separately

Now, integrate each term one by one: \[ \int (x^4 + 2x^2 + 1) \, dx = \int x^4 \, dx + \int 2x^2 \, dx + \int 1 \, dx \].

Step 4: Solve Each Integral

1. \( \int x^4 \, dx = \frac{x^5}{5} + C_1 \)2. \( \int 2x^2 \, dx = 2 \cdot \frac{x^3}{3} = \frac{2x^3}{3} + C_2 \)3. \( \int 1 \, dx = x + C_3 \).

Step 5: Combine the Results

Recombine the results of the individual integrals and simplify:\[ \frac{x^5}{5} + \frac{2x^3}{3} + x + C \], where \(C = C_1 + C_2 + C_3 \) is the constant of integration.

Key Concepts

Polynomial Expansion

When dealing with integrals, sometimes it is beneficial to expand the inner expression before integrating. In this exercise, we begin by expanding the polynomial \((x^2 + 1)^2\). This uses the formula for the expansion of a binomial expression defined by \((a + b)^2 = a^2 + 2ab + b^2\).

Here's a breakdown of this specific expansion:

• \((x^2)^2 = x^4\): This is the square of the first term.
• \(2(x^2)(1) = 2x^2\): Notice the 2, which comes from doubling the product of the two terms.
• \(1^2 = 1\): This is simply the square of the second term.

After expanding, you have the polynomial \(x^4 + 2x^2 + 1\), which is much simpler to integrate term-by-term.

Power Rule for Integration

The power rule for integration is one of the most frequently used tools for solving polynomial integrals. It states that to integrate \(x^n\), you increase the exponent by 1 and divide by the new exponent:
\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]

Let's apply this rule to each term of the polynomial we expanded:

• \(\int x^4 \, dx\): Using the power rule, this becomes \(\frac{x^5}{5} + C_1\).
• \(\int 2x^2 \, dx\): Here, factor out the 2 and use the power rule: \(2 \cdot \frac{x^3}{3} = \frac{2x^3}{3} + C_2\).
• \(\int 1 \, dx = x + C_3\)

By applying the power rule, you systematically solve each segment of the polynomial.

Step-by-Step Integration

Step-by-step integration can help simplify the problem-solving process, making sure all terms are addressed accurately. Here, we've broken it into several clear steps.

Step 1: Identify that the integral involves a polynomial expression likely benefiting from expansion rather than direct integration.
Step 2: Expand \((x^2 + 1)^2\) into a straightforward polynomial: \((x^2)^2 + 2(x^2)(1) + 1^2 = x^4 + 2x^2 + 1\).
Step 3: Integrate each term separately:

• Start with the highest power: \(\int x^4 \, dx\)
• Then move to the next: \(\int 2x^2 \, dx\)
• And finish with \(\int 1 \, dx\)

Step 4: Combine the results, remembering to include the constant of integration, represented as \(C\).

Following this methodical process ensures nothing is overlooked and provides a complete solution.

Chain Rule

The chain rule isn't directly used in this particular integration, but it's crucial to recognize its relevance when dealing with derivative operations that prompted the expanded form.

In calculus, the chain rule is used for finding the derivative of compositions of functions. The essence is in recognizing that when you have a function inside another function, like \((x^2 + 1)^2\), it plays a significant role when taking derivatives, as opposed to this integral scenario.

Often, students mistakenly apply a 'reverse' chain rule to integration problems without expanding the polynomial first, leading to incorrect results like \(\frac{1}{3}(x^2 + 1)^3\).

• Chain Rule Derivative: \((f(g(x)))' = f'(g(x)) \cdot g'(x)\)
• Integration doesn’t inherently reverse this process in such straightforward terms.

Recognizing when not to use the chain rule is as important as knowing when to apply it. In this integration, we've expanded and applied the power rule instead.