Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x^{-4} \sec ^{2}\left(x^{-3}+1\right) \sqrt[5]{\tan \left(x^{-3}+1\right)} d x\) Hint: \(D_{x} \tan x=\sec ^{2} x\)

Step-by-Step Solution

Answer

\(-\frac{5}{18} \tan^{6/5}(x^{-3} + 1) + C\).

Step 1: Choose a substitution

Notice the expression inside the functions \( \sec^2 \) and \( \tan \) is \((x^{-3}+1)\). Let's use the substitution \((u = x^{-3} + 1)\). Then, we need to calculate \((du)\).

Step 2: Differentiate u with respect to x

To find \(du\), differentiate \(u = x^{-3} + 1\) with respect to \(x\): \[ u = x^{-3} + 1 \]\[ \frac{du}{dx} = -3x^{-4} \].Thus, \(du = -3x^{-4} dx\).

Step 3: Express dx in terms of du

From the differentiated expression for \(du\), we have \(du = -3x^{-4} dx\). Solving for \(dx\):\[ dx = \frac{du}{-3x^{-4}} \].

Step 4: Substitute into the integral

Replace \(x^{-4} \cdot dx\) in the original integral with the expression for \(dx\) found in Step 3:\[ \int x^{-4} \sec^2(u) \sqrt[5]{\tan(u)} \frac{du}{-3x^{-4}} = -\frac{1}{3} \int \sec^2(u) \sqrt[5]{\tan(u)} \, du \].

Step 5: Integrate with respect to u

Notice the integral \(-\frac{1}{3} \int \sec^2(u) \sqrt[5]{\tan(u)} \, du\). Use the hint that \(D_x \tan x = \sec^2 x\), which implies:\[ \int \sec^2(u) \, d(\tan(u)) = \tan(u) \]Therefore, our integral simplifies to:\[ -\frac{1}{3} \frac{5}{6} \tan^{6/5}(u) + C \].

Step 6: Substitute back in terms of x

Recall the substitution \(u = x^{-3} + 1\). Replace \(u\) in the solution with this expression:\[ -\frac{1}{3} \frac{5}{6} \tan^{6/5}(x^{-3} + 1) + C \].

Key Concepts

Method of Substitution

The method of substitution is a powerful tool in solving indefinite integrals. It allows us to simplify complex expressions by introducing a new variable, thereby making the integral easier to evaluate. In this context, substitution acts as a change of variables. For the problem at hand, we choose the substitution \( u = x^{-3} + 1 \).

• This choice is motivated by the expression inside the functions \( \sec^2 \) and \( \tan \).
• The goal is to transform the integral into a simpler form that is easier to evaluate.

Through substitution, complex integrals can be reduced to basics, which are more straightforward to solve. After substitution, we differentiate \( u \) with respect to \( x \) to help express \( dx \) in terms of \( du \). This step is crucial as it allows us to rewrite the integral completely in terms of the new variable \( u \).

Differentiation

Differentiation is a key concept in calculus that involves finding the derivative of a function. In this exercise, it is used to find the relationship between the old variable \( x \) and the new one \( u \). Specifically, after defining \( u = x^{-3} + 1 \), differentiation gives us:\[ \frac{du}{dx} = -3x^{-4} \]

• This derivative tells us how \( u \) changes with respect to small changes in \( x \).
• It enables us to solve for \( dx \) in terms of \( du \).
• This step is necessary to completely change the variables within the integral.

The derivative \( du = -3x^{-4} dx \) reveals how to replace \( dx \) in our original integral. Without this step, we wouldn't be able to successfully execute the variable change needed for integration.

Trigonometric Integrals

Trigonometric integrals often appear in calculus, especially when dealing with functions that involve sines, cosines, tangents, and their reciprocals. In our problem, the integral contains \( \sec^2(u) \) and \( \sqrt[5]{\tan(u)} \). The hint provided uses the fact:\[ D_x \tan x = \sec^2 x \]

• This relationship helps simplify the integral \( \int \sec^2(u) \sqrt[5]{\tan(u)} \, du \) as the derivative of tan leads to simplifications.
• Understanding this relationship aids in comprehending and efficiently solving trigonometric integrals.

Using trigonometric identities and derivatives helps efficiently tackle these types of integrals, simplifying complex expressions.

Calculus Problem-Solving

Calculus problem-solving can often seem daunting at first, but can be broken down into manageable steps. This exercise showcases applying an organized approach to handle intricate integrations. With the substitution \( u = x^{-3} + 1 \) and using differentiation, we converted the original problem into a simpler form:\[ -\frac{1}{3} \int \sec^2(u) \sqrt[5]{\tan(u)} \, du \]

• Breaking down the problem into these steps assists in managing complexity and ensuring every part of the expression is accounted for.
• Each transformation step follows naturally from the previous, showing how systematic reasoning applies to calculus problems.
• Returning to the original variable \( x \) by replacing \( u \) ensures that the solution is in the terms of the variable we initially started with.

Using step-by-step approaches like substitution can ease the calculus problem-solving process by transforming intricate integral problems into those that are easier to solve. This structured method not only helps solve the problem but also builds the foundation for tackling more challenging calculus exercises.