Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \int(x+1)^{2} d x $$

Step-by-Step Solution

Answer

\( \int(x+1)^2 \, dx = \frac{x^3}{3} + x^2 + x + C \).

Step 1: Expand the Integrand

First, expand the polynomial \((x+1)^2\) using the distributive property (also known as the binomial theorem for degree 2): \((x+1)^2 = x^2 + 2x + 1\). Now the integral becomes: \[ \int (x^2 + 2x + 1) \, dx \]

Step 2: Integrate Term by Term

The integral of a polynomial can be calculated by finding the integral of each term separately. We can apply the power rule for integration here:\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.Using this rule:- The integral of \(x^2\) is \(\frac{x^3}{3}\)- The integral of \(2x\) is \(x^2\)- The integral of \(1\) is \(x\)So the integral becomes:\[ \frac{x^3}{3} + x^2 + x + C \]

Step 3: Combine Integrals and Conclude

Now, combine the integrals of each term to form the final integrated function:\[ \int (x^2 + 2x + 1) \, dx = \frac{x^3}{3} + x^2 + x + C \]This is the final integrated result.

Key Concepts

Power Rule for Integration

When tackling polynomial integration, one of the most essential tools is the Power Rule for Integration. This powerful rule states that for any function of the form \(x^n\), where \(n\) is any real number except \(-1\), the integral is given by\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]This rule allows us to find antiderivatives efficiently.
When applied to the integral of a polynomial term, such as \(x^2\), you increase the power by one to get \(x^3\), and then divide by the new power, 3. Hence, the integral of \(x^2\) is \(\frac{x^3}{3} + C\).
Another example is \(2x\), where \(x\) can be seen as \(x^1\). Following the power rule, the integral becomes \(x^2\) divided by 2, which simplifies to \(x^2 + C\).
It's crucial to remember that we apply the power rule to each term separately because integration is linear. This simplifies the process significantly.

Indefinite Integration

Indefinite Integration involves finding an antiderivative for a function, which essentially means doing the reverse of differentiation. Unlike definite integration, which results in a numeric value because of specified limits, indefinite integration provides a family of functions plus a constant of integration, denoted as \(C\).
The constant \(C\) appears because the derivative of any constant is zero, implying that multiple functions could be the antiderivative of a given function. For example:

• If you integrate \(x^2\), you will get \(\frac{x^3}{3} + C\).
• For a linear term like \(2x\), you find \(x^2 + C\).
• For a constant term, such as 1, the antiderivative is simply \(x + C\).

Indefinite integration is crucial for solving differential equations and modeling physical systems where initial conditions are unknown, which makes understanding \"\(C\)\" an integral part of comprehending the entire concept.

Binomial Theorem

The Binomial Theorem is a mathematical formula for expanding expressions that are raised to any power, such as \((x + 1)^2\).
In our exercise, we dealt with \((x + 1)^2\), a relatively simple binomial expression, which can be expanded using the distributive property. \(a = x\) and \(b = 1\), and expanding gives:

• \((x + 1)^2 = x^2 + 2x + 1\).

This process is essentially a special case of applying the broader binomial theorem, which for any power \(n\) states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
With \(\binom{n}{k}\) being the binomial coefficients.
Understanding how to expand binomials is critical when integrating, as it allows for simplifying the expression into terms that can be individually integrated using rules like the Power Rule. This simplifies the process significantly, especially for polynomial expressions.