Problems based on substitution - Trigonometric substitution

Trigonometric substitution is a technique used to simplify the integration of functions involving square roots of quadratic expressions. It involves substituting a trigonometric function for a variable to transform the integral into a form that can be more easily integrated.

When to Use Trigonometric Substitution

Trigonometric substitution is particularly useful when dealing with integrals of the form:

• $\sqrt{a^2 - x^2}$
• $\sqrt{a^2 + x^2}$
• $\sqrt{x^2 - a^2}$

where $a$ is a constant.

Types of Trigonometric Substitutions

Expression Inside the Square Root	Substitution	Reason	Trigonometric Identity Used
$a^2 - x^2$	$x = a\sin(\theta)$	Converts to $a^2(1 - \sin^2(\theta))$	$\cos^2(\theta) = 1 - \sin^2(\theta)$
$a^2 + x^2$	$x = a\tan(\theta)$	Converts to $a^2(1 + \tan^2(\theta))$	$\sec^2(\theta) = 1 + \tan^2(\theta)$
$x^2 - a^2$	$x = a\sec(\theta)$	Converts to $a^2(\sec^2(\theta) - 1)$	$\tan^2(\theta) = \sec^2(\theta) - 1$

Formulas and Trigonometric Identities

Here are some useful trigonometric identities that are often used in trigonometric substitution:

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

Examples

Example 1: $\sqrt{a^2 - x^2}$

Problem: Evaluate $\int \sqrt{a^2 - x^2} \, dx$.

Solution:

1. Use the substitution $x = a\sin(\theta)$.
2. Then $dx = a\cos(\theta) d\theta$.
3. The integral becomes $\int a\cos(\theta) \sqrt{a^2 - a^2\sin^2(\theta)} \, d\theta$.
4. Simplify the square root using the identity $\cos^2(\theta) = 1 - \sin^2(\theta)$ to get $\int a\cos(\theta) \cdot a\cos(\theta) \, d\theta$.
5. The integral simplifies to $\int a^2\cos^2(\theta) \, d\theta$.
6. Use a trigonometric identity to integrate $\cos^2(\theta)$.

Example 2: $\sqrt{a^2 + x^2}$

Problem: Evaluate $\int \sqrt{a^2 + x^2} \, dx$.

Solution:

1. Use the substitution $x = a\tan(\theta)$.
2. Then $dx = a\sec^2(\theta) d\theta$.
3. The integral becomes $\int a\sec^2(\theta) \sqrt{a^2 + a^2\tan^2(\theta)} \, d\theta$.
4. Simplify the square root using the identity $\sec^2(\theta) = 1 + \tan^2(\theta)$ to get $\int a\sec^2(\theta) \cdot a\sec(\theta) \, d\theta$.
5. The integral simplifies to $\int a^2\sec^3(\theta) \, d\theta$.
6. This integral can be more challenging and may require integration by parts or a reduction formula.

Example 3: $\sqrt{x^2 - a^2}$

Problem: Evaluate $\int \sqrt{x^2 - a^2} \, dx$.

Solution:

1. Use the substitution $x = a\sec(\theta)$.
2. Then $dx = a\sec(\theta)\tan(\theta) d\theta$.
3. The integral becomes $\int a\sec(\theta)\tan(\theta) \sqrt{a^2\sec^2(\theta) - a^2} \, d\theta$.
4. Simplify the square root using the identity $\tan^2(\theta) = \sec^2(\theta) - 1$ to get $\int a\sec(\theta)\tan(\theta) \cdot a\tan(\theta) \, d\theta$.
5. The integral simplifies to $\int a^2\sec(\theta)\tan^2(\theta) \, d\theta$.
6. Use trigonometric identities to integrate $\tan^2(\theta)$.

Conclusion

Trigonometric substitution is a powerful tool for evaluating integrals involving square roots of quadratic expressions. By using appropriate substitutions and trigonometric identities, these integrals can be transformed into more manageable forms. Practice with various examples is essential to become proficient in this technique, which is commonly tested in calculus exams.