Problem 14 - University Calculus: Early Transcendentals

Question

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \frac{1}{x^{2}} \cos ^{2}\left(\frac{1}{x}\right) d x, \quad u=-\frac{1}{x}$$

Step-by-Step Solution

Answer

The integral evaluates to \( -\frac{1}{2x} + \frac{1}{4} \sin\left(\frac{2}{x}\right) + C \).

Step 1: Identify the Substitution

We are given the substitution \( u = -\frac{1}{x} \). This implies that \( du = \frac{1}{x^2} dx \). This substitution is helpful to simplify the integral.

Step 2: Substitute and Simplify

Substitute \( u = -\frac{1}{x} \) and \( du = \frac{1}{x^2} dx \) into the integral. The integral becomes \( \int \cos^2(u) \, du \).

Step 3: Use Trigonometric Identity

We know from trigonometric identities that \( \cos^2(u) = \frac{1 + \cos(2u)}{2} \). Substitute this identity into the integral to get \( \int \frac{1 + \cos(2u)}{2} du \).

Step 4: Separate and Integrate

Split the integral into two simpler integrals: \( \frac{1}{2} \int 1 \, du + \frac{1}{2} \int \cos(2u) \, du \). Integrate each part separately.

Step 5: Integrate Each Part

The integral of \( 1 \) with respect to \( u \) is \( u \), and the integral of \( \cos(2u) \) is \( \frac{1}{2} \sin(2u) \). Therefore: \[ \frac{1}{2} \times u + \frac{1}{2} \times \frac{1}{2} \sin(2u) + C \] which simplifies to \[ \frac{u}{2} + \frac{1}{4} \sin(2u) + C \].

Step 6: Resubstitute u

Replace \( u \) with \( -\frac{1}{x} \) to get the final answer in terms of \( x \): \[ -\frac{1}{2x} + \frac{1}{4} \sin\left(\frac{2}{x}\right) + C \].

Key Concepts

Substitution Method

When evaluating indefinite integrals, one powerful technique is the "substitution method." This involves replacing a complex part of the integrand with a new variable, usually noted as "\( u \)." The purpose of using this method is to transform an integral into a simpler form, making it easier to solve.

To effectively use substitution, follow these steps:

• Identify a substitution \( u = f(x) \) that simplifies the integral. This usually involves rewriting a complicating expression.
• Determine \( du \) by differentiating \( u \) with respect to \( x \), which allows you to express \( dx \) in terms of \( du \).
• Rewrite the entire integral in terms of \( u \). This step often transforms complex integrands into something more familiar.

In our original exercise, we set \( u = -\frac{1}{x} \), which simplifies the problem. The derivative \( du = \frac{1}{x^2} dx \) conveniently matches the existing parts of the integrand, allowing for a clean substitution. By doing this, the integral transforms into a function of \( u \), paving the way for simpler calculations.

Trigonometric Identities

Utilizing trigonometric identities is a crucial step when dealing with integrals involving trigonometric functions. These identities can simplify functions, making them easier to integrate.

One common trigonometric identity is \( \cos^2(u) = \frac{1 + \cos(2u)}{2} \), which transforms powers of cosine into a simpler form that includes only basic trigonometric functions.

Here is how you apply this identity:

• Use the identity to replace \( \cos^2(u) \) with \( \frac{1 + \cos(2u)}{2} \). This helps diminish the complexity by breaking down the power of cosine into more manageable terms.
• The expression is now easier to integrate since you have separated it into basic components \( 1 \) and \( \cos(2u) \).

Such transformations using trigonometric identities are vital, as they reduce integrals to compositions of simpler functions, which can be integrated separately. In the exercise, using this identity allowed the integral to be split into two straightforward parts, each of which can be tackled individually.

Integration Techniques

Indefinite integration often requires employing specific integration techniques to solve complex integrals. These techniques build on simplified expressions, following substitution and trigonometric transformations.

In working through integrals like the one in our exercise, it's vital to:

• Split the integral, when possible, into simpler parts. In our example, \( \int \frac{1 + \cos(2u)}{2} du \) was divided into two separate integrals, \( \int 1 \ du \) and \( \int \cos(2u) \ du \).
• Integrate each part using basic integration rules: for instance, the integral of \( 1 \) with respect to \( u \) gives \( u \), and \( \cos(2u) \) integrates to \( \frac{1}{2} \sin(2u) \).

Applying these techniques ensures integrals are handled systematically and accurately. Finally, after integrating, always replace the substitution variable \( u \) back with the original expression in terms of \( x \), to revert to the original context of the problem. In our final step, \( u = -\frac{1}{x} \) was substituted back to reach a solution in terms of \( x \). This rounded off the problem, delivering a clear, well-integrated function.