Question

State the method of integration you would use to evaluate the integral \(\int x^{2} \sqrt{x^{2}-1} d x .\) Why did you choose this method?

Short answer

Use trigonometric substitution: let \( x = \sec \theta \).

Step-by-step solution

Recognize the Structure of the Integrand

The integral \( \int x^2 \sqrt{x^2 - 1} \, dx \) consists of a polynomial \( x^2 \) and a square root involving a quadratic \( x^2 - 1 \). This structure suggests a trigonometric identity might simplify the expression.

Choose Trigonometric Substitution

The expression \( \sqrt{x^2 - 1} \) reminds us of \( \sqrt{\sec^2\theta - 1} = \tan\theta \), which suggests the substitution \( x = \sec\theta \). This is because for \( x^2 - 1 \) to work for \( \sec^2\theta - 1 \), \( x \) itself would need to be expressed as \( \sec \theta \).

Implement the Substitution

Substitute \( x = \sec\theta \). Then \( dx = \sec\theta \tan\theta\, d\theta \). Substitute into the integral:\[ \int x^2 \sqrt{x^2-1} \, dx = \int \sec^2\theta \cdot \tan\theta \cdot \sec\theta \tan\theta \, d\theta \] which simplifies to \[ \int \sec^3\theta \tan^2\theta \, d\theta. \]

Justify the Choice

Choosing trigonometric substitution simplifies the integrand into expressions involving \( \sec\theta \) and \( \tan\theta \), which are easier to integrate than the original form. This transformation allows the use of trigonometric identities and further simplification.

Key concepts

Integration Techniques

Integration can become complex when dealing with certain functions. In the case of \(\int x^{2} \sqrt{x^{2}-1} \, dx\), using direct integration methods like integration by parts may not seem efficient. Therefore, trigonometric substitution is a powerful technique used to simplify such integrals. It is especially useful when the integrand involves expressions like \(\sqrt{x^2 - a^2}\), \(\sqrt{a^2 - x^2}\), or \(\sqrt{x^2 + a^2}\).

Here's why trigonometric substitution is helpful:

• Transforms a complicated radical into a simpler trigonometric identity.
• Takes advantage of known integrals of trigonometric functions.
• Provides a straightforward pathway to evaluate definite integrals if limits are known.

In this exercise, we employed the substitution \(x = \sec\theta\), turning \(\sqrt{x^2 - 1}\) into \(\tan\theta\). This choice simplifies the integration process.

Trigonometric Identities

Trigonometric identities are essential when using trigonometric substitution. These identities help simplify the expressions resulting from integration.

Key identities used include:

• \(\sec^2\theta - 1 = \tan^2\theta\)
• \(\sin^2\theta + \cos^2\theta = 1\)
• \(1 + \tan^2\theta = \sec^2\theta\)

When we substitute \(x = \sec\theta\), the integrand involves \(\tan\theta\) and \(\sec\theta\), allowing us to apply these identities. By simplifying the integral to \(\int \sec^3\theta \tan^2\theta \, d\theta\), we can use these identities to make the integration manageable. This substitution turns a seemingly complex algebraic expression into a form where calculus, involving trigonometric functions, can be easily applied.

Definite Integrals

Once an indefinite integral is solved using trigonometric substitution and identities, we can apply similar techniques to solve definite integrals. Suppose we have a specific interval \([a, b]\) for the original variable \(x\).

Here's how definite integrals work with trigonometric substitution:

• Convert the limits \(a\) and \(b\) of the integral into angles \(\theta_a\) and \(\theta_b\) using the substitution \(x = \sec\theta\).
• Evaluate the resulting integral from \(\theta_a\) to \(\theta_b\).
• Convert back to \(x\) for the final solution if necessary.

This process requires careful attention to ensure that transformations preserve the original relationship between \(x\) and \(\theta\), allowing you to accurately evaluate the definite integral with respect to the original variable.