91Ó°ÊÓ

Inverse trigonometric functions, sometimes called arc functions, reverse the processes of the usual trigonometric functions like sine, cosine, and tangent. These functions have specific properties that make them useful in calculus. In this problem, we're dealing with the inverse tangent function, written as \( \tan^{-1}(x) \). The inverse tangent function, which is arc tangent, gives the angle whose tangent is \( x \). It is important to remember that the range of \( \tan^{-1}(x) \) is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), meaning it returns angle values only within this interval.

When integrating functions involving inverse trigonometric terms, recognizing the presence can help in selecting the right techniques, like integration by parts or substitution. In this exercise, the function \( \tan^{-1}(3x) \) prompted us to leverage integration by parts to solve the integral efficiently.

Integrals are a fundamental concept in calculus, and they come in two main types: definite and indefinite. A definite integral computes the net area under a curve within specific bounds and produces a numeric value. In contrast, an indefinite integral finds a general form of the antiderivative, yielding an expression that encompasses a family of functions. Indefinite integrals come with a constant of integration, \( C \), reflecting not a single value, but rather a family of functions that differ by a constant.

In our case, evaluating \( \int \tan^{-1}(3x) \, dx \) is an example of an indefinite integral, meaning we are searching for a general expression for the antiderivative of \( \tan^{-1}(3x) \).

• Indefinite integrals don't have limits, so they include the constant of integration, \( C \).
• Finding general solutions for integrals often involves techniques like substitution and integration by parts, which allow us to tackle complex expressions that don’t directly fit standard integral forms.

The substitution method in integration is akin to reversing the chain rule in derivatives. It's a technique used to simplify an integral by making a change of variables. The core idea is to choose a substitution that makes the integral easier to evaluate.

Within this problem, substitution played a crucial role in simplifying the integral \( \int \frac{3x}{9x^2 + 1} \, dx \).

• First, we selected a substitution \( w = 9x^2 + 1 \), noting that the derivative, \( dw = 18x \, dx \), helped transform the integral into a simpler form.
• The substitution method also required expressing \( x \, dx = \frac{1}{18} dw \), allowing us to integrate with respect to \( w \) instead of \( x \).

Successfully applying the substitution method simplifies the evaluation of more complex integrals by reducing them to basic forms that are more straightforward to integrate. In our example, it transformed a potentially cumbersome integral into one involving the natural logarithm function, leading to a more manageable solution.