Problem 34 - Calculus

Question

Use the method of substitution to find each of the following indefinite integrals. $$ \int x^{-4} \sec ^{2}\left(x^{-3}+1\right) \sqrt[5]{\tan \left(x^{-3}+1\right)} d x $$

Step-by-Step Solution

Answer

Use substitution with \( u = x^{-3} + 1 \) to simplify and evaluate the integral.

Step 1: Identify the Substitution

To simplify the integral, we first identify a substitution. Notice the expression inside the trigonometric functions: \( u = x^{-3} + 1 \). This substitution simplifies the composition of functions.

Step 2: Differentiate the Substitution

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

Step 3: Solve for \( dx \) in terms of \( du \)

Rearrange the expression for \( du \) to solve for \( dx \):\( dx = -\frac{1}{3} x^4 \, du \).

Step 4: Substitute into the Integral

Substitute \( u = x^{-3} + 1 \) and \( dx = -\frac{1}{3} x^4 \, du \) into the integral:\(\int x^{-4} \sec^2(u) \sqrt[5]{\tan(u)} \left(-\frac{1}{3} x^4 \right) du\)Simplify this to:\(-\frac{1}{3} \int \sec^2(u) \sqrt[5]{\tan(u)} \, du\).

Step 5: Evaluate the Integral

Integrate the simplified integral with respect to \( u \):\(-\frac{1}{3} \int \sec^2(u) \sqrt[5]{\tan(u)} \, du\).This integral requires integration techniques beyond elementary methods, involving special functions or integrals.

Step 6: Substitute Back the Original Variable

Once the integral is evaluated in terms of \( u \), substitute back \( u = x^{-3} + 1 \) to express the answer in terms of \( x \).

Key Concepts

Trigonometric Integration

Trigonometric integration deals with integrals involving trigonometric functions like sine, cosine, or secant. These functions often appear when dealing with complex expressions, and require special techniques for integration.
The given exercise involves an integral with the trigonometric function secant squared: \( \sec^2(u) \). Here, recognizing patterns and substitutions is crucial.

• If you see expressions like \( \sec^2(u) \), it suggests a potential substitution leading to an easier integral.
• Such integrations may involve inverse trigonometric functions or logarithmic functions after substitution.

In this problem, \( \sec^2(u) \) is coupled with a radical expression like \( \sqrt[5]{\tan(u)} \). This highlights how substitutions like \( u = x^{-3} + 1 \) simplify not just the trigonometric, but also the root functions within the integral.
Understanding these foundations helps tackle more sophisticated integrals that feature similar complications.

Indefinite Integrals

Indefinite integrals, often notated as \( \int f(x) \, dx \), represent the family of all antiderivatives of a function. They are distinguished from definite integrals by not having upper or lower limits.
In practice, finding indefinite integrals means determining a general function or a collection of functions that differentiates to the given integrand.

• They include a constant of integration, denoted as \( C \), which accounts for the fact that antiderivatives of functions are not unique.
• Using substitution, a common technique for evaluating indefinite integrals, involves the change of variables to simplify the integral.

In our problem, substituting \( u = x^{-3} + 1 \) turns the integral into something more manageable. Though simplified, the resulting expression still required specific integration techniques for evaluation. After integrating in terms of \( u \), remember to substitute back to the original variable \( x \) to get the final expression of the integral.

Integration Techniques

Integration techniques are strategic methods employed for solving integrals that are not straightforward. These methods can be broadly classified into some common strategies:
The substitution method, which is prominent in our exercise, is used to simplify complex integrals by transforming variables using a substitution equation. This method effectively reduces the complexity of the integral.

• Identify a substitution that simplifies the function. For example, \( u = x^{-3} + 1 \) led to a simpler integral in our exercise.
• Compute and replace \( dx \) in terms of \( du \) to facilitate the transformation.

Substitution often works hand-in-hand with other techniques like integration by parts, partial fraction decomposition, or recognizing standard integral forms. In more complex scenarios, it could involve tables of integrals or special functions.
While simple substitutions can handle many cases, like transforming \( \sec^2(u) \) integrations, some expressions, especially those involving radicals or non-standard forms, may necessitate advanced techniques. These techniques expand the toolkit necessary for solving difficult integrals, integral to progressing in calculus.