90–103. Indefinite integrals Determine the following indefinite integrals.


∫ (2x +1)² dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves a polynomial expression squared, (2x + 1)². Expand this expression using the formula (a + b)² = a² + 2ab + b².
2. Step 2: Expand (2x + 1)² to get 4x² + 4x + 1. Rewrite the integral as ∫ (4x² + 4x + 1) dx.
3. Step 3: Break the integral into separate terms: ∫ 4x² dx + ∫ 4x dx + ∫ 1 dx. This allows you to integrate each term individually.
4. Step 4: Apply the power rule for integration to each term. For ∫ xⁿ dx, the result is (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration. Use this rule to integrate 4x², 4x, and 1.
5. Step 5: Combine the results of the individual integrals and include the constant of integration, C, to express the final indefinite integral.

Key Concepts

Indefinite Integral

An indefinite integral represents a family of functions whose derivative is the integrand. It is denoted by the integral sign followed by the function and 'dx', indicating integration with respect to x. The result includes a constant of integration (C) since the derivative of a constant is zero, meaning multiple functions can yield the same derivative.

Polynomial Expansion

Polynomial expansion involves rewriting a polynomial expression in a simplified form, often using the binomial theorem. For example, expanding (2x + 1)² results in 4x² + 4x + 1. This step is crucial for integrating polynomials, as it allows for easier application of integration rules.

Power Rule of Integration

The power rule of integration states that the integral of x raised to the power n is (x^(n+1))/(n+1) + C, where n ≠ -1. This rule simplifies the process of finding indefinite integrals of polynomial functions, making it essential for solving integrals like ∫(2x + 1)² dx after expansion.