Problem 57 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. \(\int \frac{d x}{x \sqrt{4-x^{8}}}\)

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the following function: \(\int \frac{1}{x\sqrt{4-x^8}} dx\). Answer: \(-\frac{1}{4}\frac{\sqrt{1-\frac{x^8}{4}}}{x^4} + C\)

Step 1: Attempt Substitution

Let's attempt substitution first. Let \(u = x^8\) such that \(du = 8x^7 dx\). We can rewrite the given integral as: $$\int \frac{1}{x\sqrt{4-u}}\cdot \frac{du}{8x^7}$$ Unfortunately, substitution doesn't seem to simplify the integration problem. Thus, we'll proceed with trigonometric substitution.

Step 2: Trigonometric Substitution

Let's now try trigonometric substitution. Since we have a term of the form \(\sqrt{4-u^2}\), let \(x^4 = 2\sin{\theta}\). Therefore, $$x^8 = (2\sin{\theta})^2 = 4\sin^2{\theta}$$ And $$4 - x^8 = 4 - 4\sin^2{\theta} = 4\cos^2{\theta}$$ Taking the derivative of \(x^4 = 2\sin{\theta}\), we get: $$4x^3 dx = 2\cos{\theta} d\theta \Rightarrow dx = \frac{\cos{\theta}}{2x^3} d\theta$$ Now we can substitute these expressions into our integral to simplify it: $$\int \frac{1}{x\sqrt{4-x^8}}dx = \int \frac{1}{x\sqrt{4(1-\sin^2{\theta})}} \cdot \frac{\cos{\theta}}{2x^3} d\theta$$

Step 3: Cancel out terms and simplify

Cancel out \(x\) from the numerator and denominator: $$\int \frac{1}{x\sqrt{4(1-\sin^2{\theta})}} \cdot \frac{\cos{\theta}}{2x^3} d\theta = \int \frac{1}{2x^2\sqrt{4\cos^2{\theta}}} \cdot \cos{\theta} d\theta$$ Now we can simplify the integral: $$\frac{1}{8}\int \frac{\cos{\theta}}{\sin^2{\theta}\sqrt{\cos^2{\theta}}}d\theta = \frac{1}{8}\int \frac{1}{\sin^2{\theta}} d\theta$$

Step 4: Rewrite in terms of cotangent and integrate

Using trigonometric identities, we can rewrite the integral as follows: $$\frac{1}{8}\int \frac{1}{\sin^2{\theta}} d\theta = \frac{1}{8}\int \csc^2{\theta} d\theta$$ Now we can integrate: $$\frac{1}{8}\int \csc^2{\theta} d\theta = -\frac{1}{8}\cot{\theta} + C$$

Step 5: Substitute back to original variables

Now we need to express our result in terms of \(x\). Recall that we used the substitution \(x^4 = 2\sin{\theta}\), $$\sin{\theta} = \frac{x^4}{2}$$ We can express the cotangent in terms of sine as well: $$\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} = \frac{\sqrt{1-\sin^2{\theta}}}{\sin{\theta}} = \frac{\sqrt{1-\frac{x^8}{4}}}{\frac{x^4}{2}}$$ Now substitute this back into the integral: $$-\frac{1}{8}\cot{\theta} + C = -\frac{1}{8} \cdot \frac{2\sqrt{1-\frac{x^8}{4}}}{x^4} + C$$ Finally, the indefinite integral is: $$-\frac{1}{4}\frac{\sqrt{1-\frac{x^8}{4}}}{x^4} + C$$

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals that involve root expressions. It's particularly useful when faced with expressions like \( \sqrt{a^2 - x^2} \), \( \sqrt{x^2 - a^2} \), or \( \sqrt{a^2 + x^2} \). These forms closely resemble the sides of a right triangle, allowing us to leverage basic trigonometric identities to transform the integral into a more manageable form.

In the given integral \( \int \frac{d x}{x \sqrt{4-x^{8}}} \), we encounter \( \sqrt{4 - x^8} \), a square root of a difference. We can apply trigonometric substitution by defining a variable substitution that turns this difference into a trigonometric identity. Here, by setting \( x^4 = 2\sin{\theta} \), the integral expression neatly converts the root into a cosine term: \( 4 - x^8 = 4\cos^2{\theta} \).

This substitution effectively changes the variable from \( x \) to \( \theta \), dramatically simplifying the expression and making integration feasible. Trigonometric identities come into play, transforming complicated algebraic expressions involving square roots into basic trigonometric functions.

Substitution Method

The substitution method is one of the most fundamental techniques for solving integrals. It is very similar to the reverse of the chain rule for differentiation. The main idea is to replace a complicated part of the integral with a single variable to simplify the integration process.

Initially, in the exercise, a simple substitution attempt was made: let \( u = x^8 \) and \( du = 8x^7 dx \). This process involves transforming the variable \( x \) into a new variable \( u \), simplifying the derivative expression. Unfortunately, in this case, it didn't provide an easier form to integrate, indicating that straightforward substitution isn't always directly effective.

However, don't give up on substitution easily. It's a versatile tool and sometimes, with further algebraic manipulation or different approaches, it can turn difficult integrals into standard forms that are much easier to evaluate. In this problem, substitution set the stage for the more complex trigonometric substitution that followed.

Integral Simplification

Integral simplification is the process of converting a challenging integral into a simpler, more easily solvable form. This is often achieved through algebraic manipulations, substitutions, or recognizing specific integral forms that have known solutions.

In our exercise, after applying the trigonometric substitution, simplification involved cancelling unnecessary terms and applying trigonometric identities. The use of \( \int \csc^2{\theta} d\theta \) is a clear example where recognizing a standard integral form immediately simplifies our task.

Further simplification occurred by expressing \( \cot{\theta} \) in terms of \( x \) when back-substituting. During this step, knowing the connection between inverse trigonometric functions and algebraic expressions lets us express the solution in its simplest form, ready for interpretation in terms of the original variables.

In summary, integral simplification bridges the gap between trigonometric or direct substitutions and finding the final integral in terms of the original variable. It requires a combination of algebraic know-how and familiarity with common integral forms and their solutions.