Problem 67 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \sec ^{2}(3 x) d x $$

Step-by-Step Solution

Answer

\( \int \sec^2(3x) \, dx = \frac{1}{3} \tan(3x) + C \).

Step 1: Identify the Basic Integral Form

Recognize that the integral \( \int \sec^2(u) \, du \) is a basic integrals form, which is known to equal \( \tan(u) + C \) where \( C \) is the constant of integration.

Step 2: Determine the Substitution

To use the standard integral form, set \( u = 3x \). Hence, \( du = 3 \, dx \), which implies \( dx = \frac{1}{3} \, du \).

Step 3: Substitute into the Integral

Substitute \( u \) and \( dx \) from Step 2 into the integral: \[ \int \sec^2(3x) \, dx = \int \sec^2(u) \cdot \frac{1}{3} \, du = \frac{1}{3} \int \sec^2(u) \, du. \]

Step 4: Integrate Using the Basic Integral Form

Integrate \( \frac{1}{3} \int \sec^2(u) \, du \) to obtain \( \frac{1}{3} \tan(u) + C \).

Step 5: Substitute Back the Original Variable

Re-substitute \( u = 3x \) back into the result to get \( \frac{1}{3} \tan(3x) + C \).

Key Concepts

Integration by Substitution

Integration by substitution is a fundamental technique used in calculus to solve integrals when a direct approach is not feasible. It's similar to solving a complex puzzle by transforming it into a simpler one. This concept can be thought of as the reverse chain rule.

Here is how it generally works:

• Identify a part of the integral that you can substitute with a single variable, often denoted by u. In the integration process, a function and its derivative appear, which hints at the possibility of substitution.
• Determine the differential of the new variable, typically in the form of du.
• Rewrite the original integral entirely in terms of the new variable u and du.

In our example, we noted that u = 3x. The corresponding differential is du = 3 dx, leading us to find dx = \( \frac{1}{3} du \). By substituting these into the original integral, we simplify the integration process, making the integral easier to evaluate.

Basic Integral Forms

Basic integral forms refer to the standard integration formulas that are typically memorized because they frequently appear in calculus problems. These are akin to known equations that provide direct solutions to specific forms of functions.

In our example, the integral \( \int \sec^2(u) \, du \) is recognized as one of these basic forms. Its solution is known to be \( \tan(u) + C \), where \( C \) represents the constant of integration.

• This knowledge allows us to apply the basic integral form directly after substitution.
• There are many similar basic forms involving trigonometric, logarithmic, and exponential functions.

By leveraging these known equations, calculus problems can often be solved more efficiently by reducing them to their basic components.

Calculus Problem Solving

Solving calculus problems, specifically integration problems, requires a strategic approach involving various steps.

Here's a general framework:

• First, identify any familiar forms or patterns in the integrand. This includes recognizing basic integral forms or opportunities for substitution.
• Apply integration techniques like substitution, parts, or partial fractions where applicable. For substitution, carefully choose a substitution that simplifies the problem.
• Transform the integral into a simpler form and solve using known formulas or methods.
• Always remember to substitute back the original variable to express the final solution in terms of the original integrand.

In practice, this approach often involves combining several integration techniques to tackle complex problems. As demonstrated, by following these steps, we effectively found the solution to the integral \( \int \sec^2(3x) \, dx \) as \( \frac{1}{3} \tan(3x) + C \). This structured approach is what makes calculus an invaluable tool in mathematics.