Trig substitution with tangent

TL;DR

Learn how to use trigonometric substitution to solve integrals involving the pattern a^2 + x^2, using the example of indefinite integral 1/(9 + x^2).

Transcript

Let's see if we can evaluate the indefinite integral 1 over plus 9 plus x squared dx. And we know that if you have the pattern a squared minus x squared, it could be a good idea to make the substitution, x is equal to a sine theta. But we don't see that pattern over here. Instead, what we see is a squared plus x squared. And in this context, it ten... Read More

Key Insights

• ❓ Trigonometric substitution is a powerful technique for simplifying integrals with specific patterns.
• ☺️ The substitution x = a tangent(theta) is commonly used for the pattern a^2 + x^2.
• ❓ Trigonometric substitution can be a helpful approach when other techniques fail to solve integrals.
• 💄 Making the correct trigonometric substitution helps in simplifying the integral and finding its solution.
• 😑 Trigonometric substitution can involve manipulating trigonometric identities to simplify expressions.
• ❓ The inverse trigonometric functions, such as arctan and arcsin, are often used in trigonometric substitution.
• 🙈 Trigonometric substitution can be seen as a problem-solving strategy when dealing with complicated integrals.

Questions & Answers

Q: How can trigonometric substitution be used to simplify integrals?

Trigonometric substitution is a technique that involves making a substitution of variables in an integral, using trigonometric functions. This can help simplify the integral and make it easier to solve.

Q: What is the specific trigonometric substitution used for the pattern a^2 + x^2?

For the pattern a^2 + x^2, the substitution x = a tangent(theta) can be made. This substitution simplifies the integral to a^2 sec^2(theta), which can then be integrated easily.

Q: When should one consider using trigonometric substitution?

Trigonometric substitution should be considered when other techniques like u-substitution don't work or when dealing with integrals that involve specific patterns like a^2 - x^2 or a^2 + x^2. It may not always make the integral solvable, but it is worth trying.

Q: Can you explain how the integral 1/(9 + x^2) is solved using trigonometric substitution?

To solve the integral 1/(9 + x^2), we notice that it can be rewritten as 1/(3^2 + x^2). Making the substitution x = 3 tangent(theta), we get dx = 3 sec^2(theta) d(theta). Substituting these values and simplifying, we eventually obtain the solution as 1/3 arctan(x/3) + c.

Summary & Key Takeaways

• Trigonometric substitution is a technique used to simplify integrals with specific patterns, such as a^2 - x^2 or a^2 + x^2.

• By making the substitution x = a tangent theta for the pattern a^2 + x^2, the integral can be simplified to a^2 sec^2(theta), leading to the solution.

• Trigonometric substitution can be a useful approach when other techniques like u-substitution don't work.