Problem 9 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Integrals of the Form (a) \(\int \frac{d x}{a x^{2}+b x+c}\) (b) \(\int \frac{d x}{\sqrt{a x^{2}+b x+c}}\) (c) \(\int \sqrt{a x^{2}+b x+c} d x\)Working Rule (i) Make the cocllicient of \(x^{2}\) unity by taking the coclficicnt of \(x^{2}\) outside the quadratic. (ii) Complete the square in the terms involving \(x\), i. \(.\), write \(\alpha x^{2}+b x+c\) in the form \(a[x \pm\) \(\left.\alpha)^{2} \pm \beta^{3}\right]\) (iii) The integrand is converted to one of the nine special integrals. (iv) Integrate the function.

Step-by-Step Solution

Answer

1. Make \(x^2\) coefficient unity, 2. Complete the square, 3. Identify special integral forms, 4. Substitute if necessary, 5. Integrate.

Step 1: Transform the Coefficient of x²

For each of the integrals, first make the coefficient of \(x^2\) unity. Extract the coefficient \(a\) and factor it outside the quadratic expression so that it takes the form \(x^2 + \frac{b}{a}x + \frac{c}{a}\). This transformation is important for making further manipulations simpler.

Step 2: Complete the Square

Convert the expression \(x^2 + \frac{b}{a}x + \frac{c}{a}\) into a perfect square trinomial. This involves writing it as \((x + h)^2 + k\) where \(h\) and \(k\) are constants. You can find \(h\) by using \(h = \frac{b}{2a}\) and calculate \(k\) accordingly.

Step 3: Identify the Integral Form

Once you've completed the square, the integral should resemble one of the standard forms of an integral: either \(\int \frac{dx}{(x - h)^2 + k}\), \(\int \frac{dx}{\sqrt{(x - h)^2 + k}}\), or \(\int \sqrt{(x - h)^2 + k} \, dx\). Identify which form your integral matches.

Step 4: Apply Substitution if Necessary

Depending on the form you identified, determine if a substitution is needed. Common substitutions for quadratics include setting \(u = x - h\) to simplify the integrals.

Step 5: Perform the Integration

Using the form identified in Step 3, solve the integral using appropriate integration techniques such as trigonometric substitution, partial fraction decomposition, or other calculus methods.

Key Concepts

Completing the Square

Completing the square is a technique used to simplify quadratic expressions, making them easier to work with in equations and integrals. It's especially useful when transforming a standard quadratic form into one of the perfect square forms. Essentially, you rewrite the quadratic expression into a perfect square trinomial, which is any expression of the form

• \((x + h)^2 + k\) or \((x - h)^2 + k\),

where \(h\) and \(k\) are constants.
To complete the square for an expression like \(x^2 + \frac{b}{a}x + \frac{c}{a}\), you follow a step-by-step process:

• Calculate the value of \(h\) using \(h = \frac{b}{2a}\). This value helps rewrite the linear term as \(2h(x + h)\).
• Create the perfect square trinomial by adjusting constant terms: \((x + h)^2\) and subtract any extra constant you need to balance back to the original quadratic.

By doing this, you transform a seemingly complex function into an equivalent expression that is much easier to deal with during integration.

Quadratic Expressions

Quadratic expressions play a pivotal role in calculus, especially when dealing with integrals. They are polynomial expressions of degree two. A general quadratic expression looks like this:

• \(ax^2 + bx + c\),

where \(a\), \(b\), and \(c\) are constants.
When integrating functions that contain quadratic expressions, it helps to first transform the quadratic so that the coefficient of \(x^2\) is one. This can be done by factoring out \(a\) from the entire expression. After achieving a unity coefficient, the expression becomes easier to manage and integrate.
Understanding the nature of quadratic expressions is crucial since they often require techniques like completing the square to rewrite them into forms compatible with standard integrals. By mastering how to manipulate these expressions, you can greatly simplify complex calculus problems.

Standard Integral Forms

Standard integral forms are essential tools in calculus. They encompass a group of integrals that have known solutions and are used frequently during integration.
Some of these include:

• \(\int \frac{dx}{(x - h)^2 + k}\) – This represents a form suitable for arctangent solutions.
• \(\int \frac{dx}{\sqrt{(x - h)^2 + k}}\) – This leads to arcsine or basic radical solutions.
• \(\int \sqrt{(x - h)^2 + k} \, dx\) – Typically results in integration involving exponential or logarithmic solutions.

By matching your integral to one of these forms, you simplify the process of finding an antiderivative. Recognizing these patterns through practice allows you to approach integration with confidence, knowing you have a direct path to solution by using these well-known integrals.

Substitution Method

The substitution method is like a trusty tool in the mathematician's toolkit. It's incredibly useful when faced with integrals that seem complicated or do not directly match a standard form. The technique involves replacing part of the integral with a new variable, making the problem more straightforward.
Here's how you typically use substitution:

• Identify a part of the integral that you can set as a new variable, \(u\). This is often an expression inside a square, radical, or involving multiplication with the differential \(dx\).
• Express \(dx\) in terms of \(du\) to conduct a full substitution process. This normally involves taking the derivative of \(u\) with respect to \(x\).
• Rewrite the integral in terms of \(u\). Solve the integral in this new, simplified form, and then substitute back the original expressions for your final answer.

Substitution can turn "impossible" integrals into solvable ones by simplifying complicated expressions and allowing direct use of standard integral forms. It's especially helpful for introducing new approaches in problem-solving, making seemingly complex integrals approachable and solvable.