Question

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. $$ \int \sqrt{x^{6}-x^{8}} d x $$

Short answer

Use trigonometric substitution with \( x = \sin \theta \), then simplify and integrate.

Step-by-step solution

Simplify the Expression Inside the Square Root

First, rewrite the expression inside the square root. Observe that:\[ x^{6} - x^{8} \]can be factored as:\[ x^{6}(1 - x^{2}) \]So the integral becomes:\[ \int \sqrt{x^{6}(1-x^{2})} \, dx = \int x^{3}\sqrt{1-x^2} \, dx \]

Use Trigonometric Substitution

For \( \sqrt{1-x^2} \), we use the trigonometric substitution \( x = \sin \theta \), which gives \( \sqrt{1-x^2} = \cos \theta \). Therefore, \( dx = \cos \theta \, d\theta \) and our integral becomes:\[ \int (\sin \theta)^3 \cos \theta \cdot \cos \theta \, d\theta = \int \sin^3 \theta \cos^2 \theta \, d\theta \]

Simplify Using Trigonometric Identities

Use the identity \( \cos^2 \theta = 1 - \sin^2 \theta \) to change the integral:\[ \int \sin^3 \theta (1 - \sin^2 \theta) \, d\theta = \int (\sin^3 \theta - \sin^5 \theta) \, d\theta \]

Separate and Integrate Each Term

Rewrite and integrate each term separately:\[ \int \sin^3 \theta \, d\theta - \int \sin^5 \theta \, d\theta \]First, for \( \int \sin^3 \theta \, d\theta \), use the reduction formula or power reduction formulas. A useful approach can be:\[ \int (\sin\theta)^n \, d\theta = -\frac{1}{n}(\sin\theta)^{n-1}\cos\theta + \frac{n-1}{n}\int (\sin\theta)^{n-2}\, d\theta \]Apply this to simplify and integrate both \( \int \sin^3 \theta \, d\theta \) and \( \int \sin^5 \theta \, d\theta \).

Evaluate the Integrals

Following the application of the reduction formula and calculations, find\[ \int \sin^3 \theta \, d\theta = -\frac{1}{3}(\sin\theta)^2\cos\theta + \frac{2}{3}\theta \]and similar computations for \( \int \sin^5 \theta \, d\theta \), which involve further simplifications.After solving, combine results to have:\[ F(\theta) - G(\theta) \]

Substitute Back \( \theta \) to \( x \)

Substitute back using \( x = \sin \theta \) and thus \( \theta = \arcsin x \) then replace \( \sin \theta \) and \( \cos \theta \) in terms of \( x \). The substitutions remake our antiderivative in terms of \( x \), using the derived expressions from Step 5.Express \( \cos \theta \) as \( \sqrt{1-x^2} \). The full expression resolves back into the original variable, giving a final integrated form involved with x.

Key concepts

Integration Techniques

Integrating complex expressions can often be simplified through clever techniques. One useful approach is trigonometric substitution, which exploits the known identities and properties of trigonometric functions to simplify an integral. This technique is particularly helpful when integrating expressions with square roots of quadratic polynomials.

For example, in the exercise, the integral \( \int \sqrt{x^6 - x^8} \, dx \) can be transformed using trigonometric substitution. By recognizing the expression under the square root can be simplified to \( x^3\sqrt{1-x^2} \), we can then substitute with trigonometric functions to ease integration.

Trigonometric substitution typically involves choosing a substitution that converts the radical into a simpler trigonometric function. In this example, using \( x = \sin \theta \) simplifies the expression further because \( \sqrt{1-x^2} \) turns into \( \cos \theta \). Through this substitution, the problem morphs into an integral involving trigonometric identities, which are more manageable to solve.

Trigonometric Identities

Trigonometric identities play a crucial role in the substitution method and in simplifying integrals. When \(x = \sin \theta \), we replace \(\sqrt{1-x^2} \) with \(\cos \theta \), thanks to the Pythagorean identity \(\cos^2 \theta = 1 - \sin^2 \theta\).

These identities can often transform a complex integral into a simpler form. In the given example, after substitution, we arrive at \[ \int \sin^3 \theta \cos^2 \theta \, d\theta \] which becomes \[ \int \sin^3 \theta (1-\sin^2 \theta) \, d\theta = \int (\sin^3 \theta - \sin^5 \theta) \, d\theta \.\]

This use of trigonometric identities allows us to split the integral into smaller parts, making them easier to integrate individually. In this manner, understanding and utilizing trigonometric identities is pivotal for solving integrals that initially seem daunting.

Reduction Formula for Integrals

A reduction formula simplifies the process of integrating powers of trigonometric functions. It allows us to break down a complex integral into simpler parts that can be more easily solved. For instance, with trigonometric substitution in the integral \(\int \sin^n \theta \, d\theta\), we can apply the reduction formula:

\[ \int (\sin \theta)^n \, d\theta = -\frac{1}{n}(\sin \theta)^{n-1}\cos \theta + \frac{n-1}{n}\int (\sin \theta)^{n-2} \, d\theta \]

This formula helps by expressing higher powers of sine in terms of lower powers, thus simplifying the integration process step by step. In our exercise, applying the reduction formula to \(\int \sin^3 \theta \, d\theta\) and \(\int \sin^5 \theta \, d\theta\) reduces the powers gradually, making the integration task feasible.

By sequentially applying the reduction formula, the terms become manageable and lead to a structured solution, eventually aiding in the back-substitution process to express the result in terms of the original variable \(x\). Understanding and leveraging these formulas are essential when tackling integrals involving powers of sine or cosine functions.