Problem 23 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Compute the following integrals. $$ \int \frac{\sec ^{2} \sqrt{x} \tan ^{2} \sqrt{x}}{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

\(\frac{2}{3}(2\sec^{6}(\sqrt{x}) - \sec^{3}(\sqrt{x})) + C\)

Step 1: Applying the trigonometric identity.

The integral \(\int \frac{\sec ^{2} \sqrt{x} \tan ^{2} \sqrt{x}}{\sqrt{x}} dx\) can be written as \(\int \frac{\sec ^{2} \sqrt{x} (\sec ^{2} \sqrt{x} - 1)}{\sqrt{x}} dx\) after applying the identity \(1 + \tan ^{2}(\sqrt{x}) = \sec ^{2}(\sqrt{x})\). The integral then simplifies to \(\int (\sqrt{x}\sec ^{4} \sqrt{x} - \frac{\sec ^{2} \sqrt{x}}{\sqrt{x}}) dx\).

Step 2: Applying u-substitution.

Let \(u = \sec(\sqrt{x})\). Thus, \(du = \sec(\sqrt{x})\tan(\sqrt{x}) \frac{1}{2\sqrt{x}} dx = \frac{\sec^{2}(\sqrt{x})}{2\sqrt{x}}dx\). Replacing dx in the integrals gives \(\int(2u^4 \, - u)2udu = \int(4u^5 - 2u^2)du\), which is much easier to integrate.

Step 3: Integrating and replacing u.

We can integrate this directly using the power rule for integration. The integral evaluates to \((4/6)u^6 - (2/3)u^3 + C = \frac{2}{3}(2u^6 - u^3) + C\). Replacing \(u = \sec(\sqrt{x})\), the final answer becomes \(\frac{2}{3}(2\sec^{6}(\sqrt{x}) - \sec^{3}(\sqrt{x})) + C\).

Key Concepts

u-substitution

U-substitution is a vital technique in calculus for simplifying complex integrals. It's especially helpful when dealingwith functions that compose others, like trigonometric functions or radical expressions. In our exercise, a good choice makes the integral easier by transforming it into a simpler form. Here’s how you approach it:

• Select substitution: Generally, identify a function within the integrand, such as the derivative or part of a composite expression, to set as your new variable, \(u\).
• Find \(du\): Differentiate the chosen \(u\) with respect to \(x\) and solve for \(dx\).
• Substitute: Replace all occurrences of this newly defined \(u\) into the original integral.
• Solve: Once the integral is in terms of \(u\), perform the integration.
• Back-substitute: Finally, convert back to the original variable \(x\) by replacing \(u\) with its equivalent expression in terms of \(x\).

U-substitution transforms complex integrals into manageable forms, enabling straightforward integration.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variable. They are extremely useful for manipulating and simplifying expressions during integration. A commonly used identity is the Pythagorean identity:

• \(1 + \tan^2(\theta) = \sec^2(\theta)\)

In our exercise, this identity helps to soften the complexity of the integral by converting the tangent function into a secant function. By recognizing these identities, we can rearrange terms to simplify the integral, which is crucial for performing further operations like u-substitution. The trick is to look for parts of the integral where these identities apply and use them to rewrite the integral in a more manageable form.

power rule for integration

The power rule for integration is one of the basic rules in calculus, offering a straightforward way to integrate functions that are simple power functions of the form \(x^n\). The rule states:

If \(n eq -1\), then \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\).

In the context of our example, once u-substitution has simplified the integral to a polynomial form, the power rule makes the integration process straightforward and efficient. By reducing the original integral to a sum of terms in \(u^n\), we apply the power rule to each term:

• Calculate the integral of each term individually.
• Don’t forget to add the constant of integration \(C\) at the end.

The power rule offers a simple path to finding antiderivatives, but it's important to ensure all steps like substitution and simplification accurately lead to a situation where the power rule can be directly applied.