Question

In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int \sec ^{4} x d x, \quad \sec ^{2} x=1+\tan ^{2} x$$

Short answer

\(\int \sec^4 x dx = \tan x + \frac{2}{3} \tan^3 x + \frac{1}{5} \tan^5 x + C\)

Step-by-step solution

Set Up the \(u\)-Substitution

Choose the substitution \(u = \tan x\). Then, the derivative of \(u\) with respect to \(x\) is \(\frac{du}{dx} = \sec^2 x\). Therefore, \(dx = \frac{du}{\sec^2 x}\).

Rewrite the Integral

The integral can be rewritten using the \(u\)-substitution and the trigonometric identity as: \(\int \sec^4 x dx = \int (1+u^2)^2 \frac{du}{1+u^2} = \int (1+2u^2+u^4) du\).

Evaluate the Integral

This integral can now be evaluated as a simple polynomial: \(\int (1+2u^2+u^4) du = u + \frac{2}{3} u^3 + \frac{1}{5} u^5 + C\).

Substitute Back the Value of \(u\)

Finally, substitute back in for \(u = \tan x\) to get the answer in terms of \(x\): \(\int \sec^4 x dx = \tan x + \frac{2}{3} \tan^3 x + \frac{1}{5} \tan^5 x + C\). This is the final answer.

Key concepts

U-Substitution

Understanding u-substitution is a milestone for students learning calculus. It’s a technique used to simplify complex integrals where direct integration is challenging. Think of it as an algebraic trick, similar to a magician's sleight of hand, which makes the difficult suddenly manageable.

Here's how it works: you select a part of the integral's expression and replace it with a simpler variable, often denoted as 'u'. This substitution simplifies the equation. Then, by finding the derivative of 'u' in terms of the original variable (in this exercise, 'x'), you replace 'dx' with 'du'. After performing the integration using 'u', you revert back to the original variable by substituting 'u' with its initial expression.

Trigonometric Identities

Trigonometric identities are not just mathematical curiosities—they are crucial tools for solving integrals involving trigonometric functions. The integral provided in the exercise includes a secant function raised to the fourth power. Utilizing the identity \(\sec^2 x = 1 + \tan^2 x\) simplifies the integral from a seemingly intimidating expression into a polynomial, which is far easier to integrate.

Students need to be familiar with basic identities like this one to successfully manipulate and solve trigonometric integrals. These identities express trigonometric functions in terms of one another, enabling complex expressions to be rewritten in a simpler, solvable form.

Secant Functions

The secant function, represented as \(\sec x\), might not be as familiar as its cosine counterpart, but it is equally important in calculus. It is defined as the reciprocal of the cosine function, \(\sec x = \frac{1}{\cos x}\). The integral in the example increased this function's complexity by raising it to the fourth power.

Understanding the behavior and properties of secant functions is vital, especially in evaluating their integrals. In this problem, employing the trigonometric identity involving the secant function makes it possible to convert the integral into one involving tangent, which students might find more approachable.

Antiderivatives

The notion of an antiderivative is fundamental in calculus. An antiderivative of a function is basically another function that, when differentiated, returns the original function. This means the search for antiderivatives is a quest to find a function whose rate of change (derivative) is known to us.

Integrating \(1+2u^2+u^4\) with respect to \(u\) yields an antiderivative. Each term of this polynomial is integrated separately, producing a sum of power functions and a constant of integration, denoted as 'C'. This constant encapsulates all possible vertical shifts of the antiderivative, highlighting the fact that integrals represent a family of functions rather than a single solution.