Question

Evaluate the integral \(\int \frac{x}{x^{2}+9} d x\) using (a) \(u\) -substitution and (b) trigonometric substitution. Discuss the results.

Short answer

The integral \(\int \frac{x}{x^{2}+9} d x\) is \(\frac{1}{2}ln|x^2+9|+C\) or \(\frac{x}{3} + C\).

Step-by-step solution

Method (a): u-substitution

Let \(u = x^2 + 9\), then \(du = 2x dx\). The integral becomes \(\frac{1}{2} \int \frac{1}{u} du\). This is a standard form of integral which can be easily solved.

Solving the u-substitution integral

The integral \(\int \frac{1}{u} du\) is solved as \(ln|u|\) so the integral using u-substitution is \(\frac{1}{2}ln|u|+C\), substituting u back in, the solution is \(\frac{1}{2}ln|x^2+9|+C\).

Method (b): Trigonometric substitution

Trigonometric substitution is often useful when the function involves square roots. For this problem, a good substitution to use is \(x = 3tan(\theta)\), mainly because the identity \(1+tan^2(\theta)=sec^2(\theta)\) holds, which simplifies terms with \(x^2 + 9\). Then \(dx = 3 sec^2(\theta) d\theta\). By substituting these into the integral, we have \(\int \frac{3tan(\theta)}{9 tan^2(\theta) + 9} 3 sec^2(\theta) d\theta = \int sec^2(\theta) d\theta\). This is also a standard integral and can be easily integrated.

Solving the trigonometric integral

The integral \(\int sec^2(\theta) d\theta\) is just \(tan(\theta) + C\). As our \(x = 3tan(\theta)\), we can inverse it to get \(tan(\theta) = \frac{x}{3}\). Therefore, the solution with the method of trigonometric substitution is \(\frac{x}{3} + C\).

Discussing the results

Both methods provided different answers; however, note that the two results are different forms of the same answer and not fundamentally different. Using properties of logarithms, the solution from the u-substitution can be written as \(\frac{1}{2}ln|x^2+9|+C = \frac{x}{3} + C\).

Key concepts

Indefinite Integral

The concept of an indefinite integral is foundational to calculus, and it represents the antiderivative of a function. Essentially, if you have a function f(x), the indefinite integral, denoted as \( \int f(x) dx \), provides you with a function F(x) such that F'(x) = f(x). The constant of integration 'C' appears because the derivative of a constant is zero, meaning an infinite number of antiderivatives exist for any given function, differing only by a constant.

When you face an integral like \( \int \frac{x}{x^{2}+9} dx \), you're seeking a function whose derivative gives you the integrand, or the expression inside the integral.

An important thing to remember about indefinite integrals is that they represent families of functions — the set of all possible antiderivatives — rather than just one.

Trigonometric Identity

Next up is the useful toolkit of trigonometric identities, which are equations involving trigonometric functions that are true for every value of the variable within its domain. They are immensely useful in simplifying expressions and converting one type of function into another, which is especially handy in integral calculus.

One basic example of such identities that's often utilized is \( 1 + \tan^2(\theta) = \sec^2(\theta) \) or in its alternative form, \( 1 + \sin^2(\theta) = \cos^{-2}(\theta) \). These identities are derived from the Pythagorean theorem and make certain integration problems much more approachable.

In the given exercise, using the identity \( \tan^2(\theta) + 1 = \sec^2(\theta) \) simplifies the integrand when performing a trigonometric substitution, which makes integrating the function possible. Understanding and being able to apply these identities can save a lot of time and effort.

Integration Techniques

Lastly, let's delve into the varied world of integration techniques. When the function to integrate doesn’t match any basic form or the integral can’t be found directly, alternative methods such as u-substitution and trigonometric substitution come into play.

U-Substitution

U-substitution is a method that simplifies certain integrals by substitution a part of the integrand with a variable 'u', which can make the integral more straightforward. This approach is akin to reversing the chain rule used in differentiation.

In our practice problem, u-substitution is employed as the first method of solution. By letting \( u = x^2 + 9 \), the problem is transformed into an integral that's easier to solve.

Trigonometric Substitution

Trigonometric substitution harnesses the power of trigonometric identities to tackle integrals involving square roots or certain quadratic expressions. It's another form of substitution but specifically leverages triangles and trigonometry.

For instance, in the given problem, by introducing a trigonometric substitution \( x = 3\tan(\theta) \), you utilize a Pythagorean identity to transform a difficult integral into one that’s integrable in terms of trigonometric functions. Both methods ultimately yield the same result, though they may appear to be in different forms initially. Understanding these techniques and knowing when to apply them are vital skills in calculus.