Printable Practice Problems: Trigonometric Substitution for `√(a²+x²)` Integrals

📚 Topic Summary

Trigonometric substitution is a technique used to simplify integrals containing expressions of the form $\sqrt{a^2 + x^2}$, $\sqrt{a^2 - x^2}$, or $\sqrt{x^2 - a^2}$. For integrals involving $\sqrt{a^2 + x^2}$, we typically use the substitution $x = a\tan(\theta)$. This substitution allows us to eliminate the square root and simplify the integral using trigonometric identities. After evaluating the integral in terms of $\theta$, we convert back to $x$ using the initial substitution and a reference triangle.

This worksheet focuses on integrals containing the form $\sqrt{a^2 + x^2}$. By working through these problems, you'll gain confidence in applying trigonometric substitution and mastering the art of integral calculus!

🧮 Part A: Vocabulary

Match the term to its definition:

1. Substitution
2. Integral
3. Trigonometry
4. Hypotenuse
5. Tangent

1. The side opposite the right angle in a right triangle.
2. Replacing a variable with an expression.
3. The branch of mathematics dealing with relationships between angles and sides of triangles.
4. The reverse process of differentiation; finding the area under a curve.
5. In a right triangle, the ratio of the opposite side to the adjacent side.

✏️ Part B: Fill in the Blanks

For integrals containing the expression $\sqrt{a^2 + x^2}$, we use the trigonometric substitution $x = a\tan(\theta)$. This substitution helps to simplify the integral because $\sqrt{a^2 + a^2\tan^2(\theta)}$ can be simplified to $a\sec(\theta)$ using the trigonometric identity $1 + \tan^2(\theta) = \sec^2(\theta)$. After integrating, we convert back to the original variable using a _______ triangle and the initial _______.

🤔 Part C: Critical Thinking

Explain why the substitution $x = a\sin(\theta)$ is NOT suitable for integrals containing $\sqrt{a^2 + x^2}$. What type of integral IS it suitable for?