Problem 9 - Calculus Single Variable

Question

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

Step-by-Step Solution

Answer

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

Step 1: Expand the Integrand

To solve the integral, start by expanding the expression \(x + 1\)^2. This gives us: \\[(x + 1)^2 = x^2 + 2x + 1\]

Step 2: Integrate Each Term Separately

Now that we have \(x^2 + 2x + 1\), we can integrate each term separately: \- Integrate \(x^2\): \int x^2 \, dx = \frac{x^3}{3}\- Integrate \(2x\): \int 2x \, dx = x^2\- Integrate \(1\): \int 1 \, dx = x\.\Combine these results to get the integrated expression.

Step 3: Combine the Results

Combine the integrals of each term: - \frac{x^3}{3} + x^2 + x\Don't forget to add the constant of integration, \C\. Thus, the indefinite integral is \[\frac{x^3}{3} + x^2 + x + C\].

Step 4: Verify the Results

To verify, differentiate the result \[\frac{x^3}{3} + x^2 + x + C\] with respect to \(x\) and check that it matches the original expression: - \frac{d}{dx}\left(\frac{x^3}{3}\right) = x^2\ - \frac{d}{dx}(x^2) = 2x\ - \frac{d}{dx}(x) = 1\The result \(x^2 + 2x + 1\) matches the expanded form of the original integrand, confirming the solution is correct.

Key Concepts

Integration by Substitution

Integration by substitution is a powerful technique that aims to simplify integrals by making a smart substitution. It's like untangling a knot by finding a clever thread to pull. You replace a part of the integrand with a new variable, which often simplifies the integral into a form that's easier to work with.

Here’s how it works:

• Identify a part of the integrand that can be substituted with a new variable, say \( u \).
• Express \( dx \) in terms of \( du \) by differentiating your substitution.
• Rewrite the entire integral in terms of \( u \) and \( du \).
• Integrate with respect to \( u \).
• Substitute back into the original variable.

Substitution is especially useful when dealing with integrals involving composite functions or expressions that resemble the derivative of another expression.

Integration Techniques

Integration is a crucial tool in calculus, similar to finding the area under a curve. However, not all functions can be integrated directly, which is why different techniques exist to tackle various types of integrals.

Some common integration techniques include:

• Basic Integrals: These involve straightforward functions like \( x^n \), where you integrate directly using power rules.
• Integration by Parts: Useful when an integrand is the product of two functions, relying on the product rule of differentiation.
• Partial Fraction Decomposition: Helps integrate rational functions by breaking them into simpler fractions.
• Trigonometric Substitution: Used for integrals involving expressions like \( \sqrt{a^2 - x^2} \).

Knowing when and how to use each technique is key to solving complex integrals efficiently.

Expanding Expressions

Expanding an expression can significantly simplify the process of integration, as seen in our example.

• Firstly, understand the expression: What does it represent?
• Next, rewrite it in a form that makes it easier to handle. For example, \((x+1)^2\) expands into \(x^2 + 2x + 1\).
• Once expanded, you integrate each term separately.

In many cases, expansion helps remove the complexity of dealing with powers or composite expressions directly, thus allowing for simple integration term by term. Always remember, after integrating, combine all terms together and add the constant of integration \( C \). This complete expression represents the indefinite integral you seek.