Problem 37 - Applied Calculus

Question

Find the indefinite integrals. $$ \int(x+1)^{2} d x $$

Step-by-Step Solution

Answer

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

Step 1: Expand the Integrand

Start by expanding the expression \((x + 1)^2 = x^2 + 2x + 1\). This involves squaring the binomial.

Step 2: Integrate Each Term Separately

Now, integrate each term in the expanded polynomial separately:\[\int(x^2 + 2x + 1) \, dx = \int x^2 \, dx + \int 2x \, dx + \int 1 \, dx.\]

Step 3: Apply the Power Rule

Use the power rule for integration, which states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\):\[ \int x^2 \, dx = \frac{x^3}{3}, \]\[ \int 2x \, dx = 2 \cdot \frac{x^2}{2} = x^2, \]\[ \int 1 \, dx = x. \]

Step 4: Combine the Results

Combine the results from the previous integrations:\[ \frac{x^3}{3} + x^2 + x + C, \]where \(C\) is the constant of integration.

Key Concepts

Integration Techniques

Integration techniques help us find the integral of a function, which is like finding the area under a curve. When dealing with integrals, especially indefinite ones that have no set limits, different strategies apply. There isn't just one way to find these integrals; different techniques are suitable for different types of problems. For the case of \( \int(x+1)^{2} \, dx \), let's break down the technique used:

• **Expand**: Expand the expression to simplify it before integration. This might involve using a technique like the binomial expansion to rewrite expressions in a simpler form.
• **Separate and Integrate**: After expanding, you integrate each term separately, making use of simpler integration rules.

These techniques can make challenging problems more approachable. Integrating simpler functions term by term after expansion is often more manageable than tackling the whole expression at once. This step-by-step approach is essential in many math problems, especially when finding indefinite integrals.

Power Rule

The power rule is a fundamental part of calculus and is one of the first techniques you encounter when learning to integrate. It's simple and very efficient for polynomial functions. The power rule states:
\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C,\]where \(n eq -1\), and \(C\) is the constant of integration.
Key points about the power rule:

• **Positive Exponents**: For any positive integer \(n\), this rule applies directly.
• **Combining Integrals**: It's often used when you have a polynomial expanded form where you just integrate each term separately using the rule.
• **Constant Multipliers**: If a term in a polynomial is multiplied by a constant, integrate the polynomial part first and then multiply by the constant afterward.

Understanding the power rule is critical because it lays the foundation for more complex integration techniques. Remember that this rule does not apply to functions like \(1/x\) directly; for those, another method called the natural logarithm integration is used. But whenever you spot a polynomial, think power rule.

Binomial Expansion

The binomial expansion is a handy algebraic tool to simplify expressions like \((x+1)^2\). This expansion lets us express powers of binomials as a sum of terms, making integration much easier.
For a binomial \((a + b)^n\), it expands according to the binomial theorem:

• **Formula Basics**: Each term is of the form \( \binom{n}{k} a^{n-k}b^k \) where \( \binom{n}{k} \) are binomial coefficients.
• **Simple Expansion**: For our exercise, \((x+1)^2 = x^2 + 2x + 1\). It's straightforward because the power is only 2.
• **Usefulness in Integration**: By breaking down the expression, you can apply integration directly to each term, avoiding complex integrations.

For small powers like two, the expansion is simple enough to do directly without formula application. However, for higher powers, remembering or using the binomial theorem is useful. This simplification is a crucial first step in many calculus problems, like our indefinite integral problem. Breaking things down into manageable pieces is key to solving complex problems smoothly.