Problem 70 - Calculus Early Transcendentals

Question

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int \sec ^{4} 4 t d t$$

Step-by-Step Solution

Answer

Question: Evaluate the following integral: $$\int \sec^4 4t dt$$ Answer: The integral can be evaluated as: $$\int \sec^4 4t dt = \frac{1}{3}(\sec^2 4t\tan 4t) + \frac{1}{12}(\tan 4t + C)$$

Step 1: Apply the reduction formula for secant

We use the reduction formula for the secant function: $$\int \sec^n x dx = \frac{1}{n-1}(\sec^{n-2} x\tan x) + \frac{n-2}{n-1} \int \sec^{n-2} x dx$$ Here, n = 4. Apply the reduction formula to $$\int \sec^4 4t dt$$: $$\int \sec^4 4t dt = \frac{1}{3}(\sec^2 4t\tan 4t) + \frac{1}{3} \int \sec^2 4t dt$$

Step 2: Perform substitution for the second integral term

Let us make a substitution for the integral in the expression: Let $$u = 4t$$ Then, $$du = 4dt$$ Hence, $$\frac{1}{4} du = dt$$ Now, substitute in the integral: $$\frac{1}{3} \int \sec^2 4t dt = \frac{1}{3} \int \sec^2 u \frac{1}{4} du = \frac{1}{12} \int \sec^2 u du$$

Step 3: Evaluate the integral

Now, we know that: $$\int \sec^2 u du = \tan u + C$$ So, we have: $$\frac{1}{12} \int \sec^2 u du = \frac{1}{12}(\tan u + C)$$ Substitute back for $$u$$ and get the final result: $$\frac{1}{12}(\tan 4t + C)$$

Step 4: Combine the results and write the final answer

Now, combine the results from Step 1 and Step 3 to write the final answer: $$\int \sec^4 4t dt = \frac{1}{3}(\sec^2 4t\tan 4t) + \frac{1}{12}(\tan 4t + C)$$

Key Concepts

Reduction Formulas

Reduction formulas are a handy tool in calculus, especially when dealing with complex integrals involving powers of functions like secant or sine. Think of them as shortcuts that simplify the integration process by reducing the power of the function step-by-step.
For the secant function, the reduction formula is:\[\int \sec^n x \, dx = \frac{1}{n-1}(\sec^{n-2} x \tan x) + \frac{n-2}{n-1} \int \sec^{n-2} x \, dx\]This formula breaks down a higher-power secant integral into a part that involves a product of functions and another integral of a lower power.

• The formula is recursive, meaning you use it repeatedly until the problem is reduced to a simpler integral.
• This technique is efficient because it systematically reduces the complexity of the integral each time it is applied.

In our example, with \(n = 4\), using the reduction formula simplifies the integration process drastically by giving a mix of a solvable integral and a ternary expression that becomes easier to handle.

Secant Function

The secant function, denoted as \( \sec x \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, defined as \( \sec x = \frac{1}{\cos x} \). This function is particularly important in calculus and trigonometry because of its distinctive properties involving asymptotes and periodicity.
Some noteworthy features of the secant function include:

• It has no range restrictions but has domain restrictions where cosine is zero because division by zero would imply undefined values.
• The typical interval for observing its behavior is any interval of size \(2\pi\), such as \([0, 2\pi)\), due to its periodic nature.

When integrated, as seen in our exercise, the secant function can create complex expressions, hence why using reduction formulas and substitutions can be highly beneficial.

Substitution Method

The substitution method is a powerful integration technique that simplifies an integral by changing variables, often transforming it into a more approachable problem. In essence, this method is akin to what is called 'u-substitution'. It works on the principle of changing the variables to match standard integral forms or reduce compounds.
Here's how it's generally applied:

• Select a portion of the integral to substitute (usually a composite function or a trigonometric expression), such that \( u = g(x) \).
• Compute \( du \), the differential of \( u \) with respect to the original variable, to eventually replace it in the integral.
• Rewrite the original integral in terms of \( u \) which often simplifies solving it.

In the provided example, the substitution \( u = 4t \) allowed a more straightforward integration of \( \sec^2 u \). It reduced the original problem involving \( t \) into an integral with \( u \) that resembles a standard form, making it easier to integrate and ultimately find the solution.