Question

Find the indefinite integral in two ways. Explain any difference in the forms of the answers. $$ \int(2 x-1)^{2} d x $$

Short answer

The indefinite integral can be expressed in two ways depending upon the method used to calculate it: \(\frac{1}{6}(2x-1)^{3} + C\) or \(\frac{4}{3}x^{3} - 2x^{2} + x + C\). Although they appear different, it's notable that these two forms are equivalent as they differ by a constant, which doesn't affect the integrity of the integral.

Step-by-step solution

Method 1: Direct Application of Power Rule

First, apply the direct power rule which states the integral of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\). Apply the substitution \(u = 2x - 1\) then differentiate \(u\) to get \(du = 2dx\) or \(dx = \frac{1}{2}du\). Substituting into the integral, it becomes \(\int u^{2} * \frac{1}{2} du\) which finally translates to \(\frac{1}{2} \int u^{2} du\), evaluated using the power rule to be \(\frac{1}{2} * \frac{1}{3} * u^{3} = \frac{1}{6}*(2x-1)^{3} + C\) where \(C\) is the constant of integration.

Method 2: Expansion Before Integration

Alternately, expand \( (2x-1)^2 \) before integrating. The expression becomes \(\int (4x^2 - 4x + 1) dx = \int 4x^2 dx - \int 4x dx + \int dx\). Now compute the integral of each term separately. The integral of \(4x^2\) is \(\frac{4}{3}x^{3}\), the integral of \(4x\) is \(2x^{2}\) and the integral of \(1\) or \(dx\) is \(x\). Combining all of these, the answer becomes \(\frac{4}{3}x^{3} - 2x^{2} + x + C\).

Explanation of Differences

Despite the different forms, these answers are equivalent because of the flexible nature of the constant of integration. The difference between them is due to the different methods of approaching the integral: one method involves direct application of the power rule with substitution, while the other starts with an expansion of the quadratic term. The difference is purely an aesthetic one due to the choice of methods.