key term - $ an^{-1}$

Definition

$ an^{-1}$ is the inverse trigonometric function of the tangent function. It represents the angle whose tangent is a given value. This term is particularly important in the context of integrals resulting in inverse trigonometric functions, as it allows for the evaluation of certain types of integrals involving tangent functions.

5 Must Know Facts For Your Next Test

1. $ an^{-1}$ is used to find the angle whose tangent is a given value, denoted as $ an^{-1}(x)$.
2. The domain of $ an^{-1}$ is the set of all real numbers, and its range is $(-rac{ ext{pi}}{2}, rac{ ext{pi}}{2})$.
3. The derivative of $ an^{-1}(x)$ is $rac{1}{1+x^2}$.
4. Integrals involving $ an^{-1}$ can be evaluated using trigonometric substitution, where $x = an(u)$.
5. The integral of $rac{1}{1+x^2}$ is $ an^{-1}(x) + C$.

Review Questions

• Explain the relationship between $ an^{-1}$ and the tangent function.
  • The $ an^{-1}$ function is the inverse of the tangent function. This means that if $y = an(x)$, then $x = an^{-1}(y)$. In other words, $ an^{-1}$ allows you to find the angle whose tangent is a given value. The domain of $ an^{-1}$ is the set of all real numbers, and its range is $(-rac{ ext{pi}}{2}, rac{ ext{pi}}{2})$, which corresponds to the angles where the tangent function is defined.
• Describe how $ an^{-1}$ is used in the context of integrals resulting in inverse trigonometric functions.
  • In the context of integrals resulting in inverse trigonometric functions, $ an^{-1}$ is used as part of a trigonometric substitution technique. When evaluating integrals involving certain trigonometric functions or expressions, such as $rac{1}{1+x^2}$, you can substitute $x = an(u)$, which then allows you to express the integral in terms of $ an^{-1}(x)$. This substitution technique helps simplify the integration process and leads to the evaluation of the integral in terms of the inverse tangent function.
• Analyze the role of the derivative of $ an^{-1}(x)$ in the context of integrals resulting in inverse trigonometric functions.
  • The derivative of $ an^{-1}(x)$ is $rac{1}{1+x^2}$. This derivative property is crucial in the context of integrals resulting in inverse trigonometric functions. When evaluating integrals involving expressions like $rac{1}{1+x^2}$, you can use the fact that the integral of $rac{1}{1+x^2}$ is $ an^{-1}(x) + C$. This connection between the derivative of $ an^{-1}(x)$ and the integral of $rac{1}{1+x^2}$ is a key concept in the evaluation of integrals resulting in inverse trigonometric functions.

Related terms

Inverse Trigonometric Functions:

Inverse trigonometric functions are the inverse of the standard trigonometric functions, such as sine, cosine, and tangent. They allow you to find the angle given the value of the trigonometric function.

Trigonometric Substitution: Trigonometric substitution is a technique used in integral calculus to evaluate integrals involving certain trigonometric functions or expressions.

Indefinite Integral: An indefinite integral is an antiderivative or primitive function of a given function. It represents the set of all functions whose derivative is the given function.