Problem 36 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{d x}{\sin ^{2} 3 x}$$

Step-by-Step Solution

Answer

-\frac{1}{3} \cot 3x + c

Step 1: Choose Appropriate Substitution

In our case, let's choose \(u = 3x\). This simplifies the integral and its subsequent steps.

Step 2: Calculate Differential of u

Differentiating \(u = 3x\) we get, \(du = 3dx\). To isolate \(dx\) we then divide both sides by 3, \(dx = \frac{du}{3}\).

Step 3: Substitute u and dx in the Integral

Now, substitute \(u\) and \(dx\) into the integral. The integral \(\int \frac{dx}{\sin^{2}3x}\) becomes \(\int \frac{1}{\sin^{2}u} * \frac{du}{3}\). We factor out constants from the integral, giving us \(\frac{1}{3} \int \frac{du}{\sin^{2}u}\)

Step 4: Evaluate the Integral

The integral \(\frac{1}{3} \int \frac{du}{\sin^{2}u}\) is a standard integral which equates to \(-\frac{1}{3} \cot u + c\), where 'c' is the constant of integration.

Step 5: Back Substitute u Into the Answer

Substituting \(u = 3x\) into \(-\frac{1}{3} \cot u + c\), we get our final answer as \(-\frac{1}{3} \cot 3x + c\).

Key Concepts

Integration

Integration is a fundamental concept in calculus. It helps to find areas under curves, among other things. Think of it as a way to accumulate quantities. Unlike differentiation, which breaks down functions into smaller parts, integration pieces them back together.
Integration can be understood in two main ways:

• Definite Integrals: These are used when you have limits and want to calculate the accumulated value between two points.
• Indefinite Integrals: These have no limits and are often represented with a constant, usually denoted as 'c'.

In our exercise, we deal with an indefinite integral because there are no specific boundaries, and we include a constant at the end.

Substitution Method

The substitution method is a powerful technique used in integration. It simplifies complex integrals by changing variables, much like solving a puzzle. You identify parts of the integral that can be substituted for simpler expressions. It's like changing the language of the problem.
Here's how it works:

• Choose a Substitution: This step involves identifying a part of the integral and setting it equal to a new variable (e.g., \(u = 3x\)).
• Differentiate: Find the derivative of your chosen substitution to express \(dx\) (e.g., \(du = 3dx\)).
• Substitute: Replace all instances of the original variable and \(dx\) with your new variable and its differential (e.g., \(dx = \frac{du}{3}\)).

This method helps transform the integral into a more manageable form.

Trigonometric Integrals

Trigonometric integrals are those that involve trigonometric functions like sine or cosine. These functions often appear in calculus due to their repetitive nature. A common strategy to solve them involves using identities or specific techniques like substitution.
In the exercise, the integral involves \(\sin^2(3x)\). Trigonometric identities can sometimes simplify these expressions further. Once you perform the substitution, the integral can appear simpler, as you replace \(\sin^2(u)\) and integrate with respect to this new variable.

Definite and Indefinite Integrals

Definite and indefinite integrals are two sides of the same coin. They both pertain to integration but serve different purposes.

• Definite Integrals: Evaluate the net area between the curve and x-axis within given limits, providing a numerical value.
• Indefinite Integrals: Represent all antiderivatives of a function. They are more general and include a constant of integration, denoted by 'c'.

In our exercise, we compute an indefinite integral by finding the antiderivative. The result comes with a +c because there’s no upper or lower limit specified. This constant accounts for all possible vertical shifts of the antiderivative.