91Ó°ÊÓ

Completing the square is a technique used in algebra and calculus to transform a quadratic expression into a perfect square trinomial.This method is especially helpful when dealing with integrals of quadratic functions, where the expression is in the form of \(ax^2 + bx + c\).
Here, our expression is \(2x^2 - 3x + 9\). To complete the square, we start by factoring out the leading coefficient of the quadratic terms. In this case, it's 2, resulting in:

• \(2(x^2 - \frac{3}{2}x) + 9\)

Next, we need to add and subtract the square of half of the coefficient of \(x\) from within the bracket. Half of \(-\frac{3}{2}\) is \(-\frac{3}{4}\). We square it to get \(\frac{9}{16}\), leading to:

• \(2\left((x - \frac{3}{4})^2 - \frac{9}{16}\right) + 9\)

Finally, simplify the expression to get it into a neat square form:

• \(2(x - \frac{3}{4})^2 + \frac{63}{8}\)

This transformation allows us to use more advanced integration techniques, making solving the integral feasible.

Trigonometric substitution is a method used to simplify integrals involving square roots of quadratics or rational expressions.Once the quadratic expression has been transformed into a square, as in the function \(2(x - \frac{3}{4})^2 + \frac{63}{8}\), trigonometric substitution is often the go-to choice if the resulting expression resembles standard trigonometric integrals.
We aim to match our integral to the \ extit{arctangent}\ form:

• \(\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \cdot \tan^{-1}{\left(\frac{x}{a}\right)} + C\)

To achieve this with the expression \(\int \frac{8}{16u^2 + 63} \, du\), set \(a^2\) to \(\frac{63}{16}\).This means \(a\) is \(\frac{\sqrt{63}}{4}\), and we can use it to rewrite the integral in terms of an arctangent function.
Thus, the integral becomes:

• \(\frac{1}{a} \cdot \tan^{-1}{\left(\frac{u}{a}\right)}\)

Substituting and simplifying gives us a neatly solved integral form that we can revert back to terms of \(x\) later on.

Indefinite integrals, unlike definite integrals, do not evaluate to a specific numerical value.They represent a family of functions and include an arbitrary constant, \(C\).The indefinite integral we are tackling here is written as:

• \(\int \frac{1}{2x^2 - 3x + 9} \, dx\)

The goal is to find an antiderivative of the function, which includes the constant \(C\) that signifies all possible vertical shifts of the resulting function.
Our approach, involving completing the square and trigonometric substitution, takes the given rational function and rewrites it so a known antiderivative can be applied, helping us integrate the expression back into terms of \(x\) with a constant added at the end.

The substitution method is a crucial technique where a substitution is made to simplify an integral.In our problem, this is done with the variable \(u\), defining it in relation to \(x\).By setting \(u = x - \frac{3}{4}\), we transform the original integral into

• \(\int \frac{1}{2u^2 + \frac{63}{8}} \, du\)

This method of substitution transforms the expression into a simpler form that is easier to integrate. The differential \(dx\) is also converted to \(du\), making the transition seamless.
Once the integral is evaluated using this substitution, the final step involves reversing the substitution, replacing \(u\) back with \(x - \frac{3}{4}\) to find the function in terms of the original variable of integration.This method can significantly streamline the complexity of evaluating an integral.