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The substitution method is a powerful technique used to simplify complex integrals. It involves changing the variable of integration to make the integral easier to solve.
The main goal of substitution is to transform a difficult integral into a simpler form. This simplifies the calculation and achieves an easier solution.When applying the substitution method, you generally follow these steps:

• Identify a part of the integrand to substitute with a new variable, usually by setting a complex expression equal to this new variable (commonly denoted as \(u\)).
• Differentiate this equation to find \(du\) in terms of \(dt\).
• Rewrite the original integral using \(u\) and \(du\) instead of \(t\) and \(dt\).

In our exercise, we chose \(u = 2 + 3t\). This choice simplifies the original integral \(\int(t+2) \sqrt{2+3t} \, dt\) into something more manageable. By differentiating, we find \(du = 3 \, dt\). Replacing \(dt\) with \(\frac{du}{3}\), the integral becomes easier to compute.

Polynomial functions are expressions that involve terms in the form of \(x^n\), where \(n\) is a non-negative integer. In integrals, polynomials can often be split and handled individually, making them easier to integrate.In our integral, the expression \(t+2\) is a polynomial of degree 1, meaning it involves \(t\) raised to the power of 0 and 1 respectively.
These types of functions are straightforward to work with when integrating. After substituting \(u = 2 + 3t\), the polynomial \(t + 2\) is rewritten in terms of \(u\), making the entire expression much simpler.Key points about polynomial functions in integration:

• Polynomials can be integrated term by term.
• They often require algebraic manipulation to fit into a solvable form after substitution.
• Each term's integration involves using the power rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) where \(n eq -1\).

In the exercise, the polynomial \(\frac{u-2}{3} + 2\) got simplified and integrated term by term, utilizing polynomial integration techniques effectively.

The square root function is characterized by its form \(\sqrt{x}\), which can introduce complexity in integration. However, when paired with substitution, it becomes much more tractable.For our integral \(\int(t+2) \sqrt{2+3t} \, dt\), the square root function \(\sqrt{2+3t}\) is reworked through substitution, letting \( u = 2 + 3t \).
The expression \(\sqrt{u}\) replaces \(\sqrt{2+3t}\), which allows the integral to be addressed easily.Integrating square roots often requires:

• Rewriting the square root in exponent form (e.g., \(\sqrt{u} = u^{1/2}\)).
• Utilizing substitution to transform complicated expressions into more straightforward forms.

In this exercise, transforming \(\sqrt{2+3t}\) through substitution to \(u^{1/2}\), we applied the power rule for integration.
This rule dictates how to handle expressions raised to a fractional power, thus making the overall integration feasible.