key term - $ extsec^{-1}$

Definition

$ extsec^{-1}$ is the inverse of the secant function, which is one of the inverse trigonometric functions. It represents the angle whose secant 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 the secant function. The inverse secant function, $ extsec^{-1}$, is a way to undo the secant function, just as the inverse sine, cosine, and tangent functions undo their respective trigonometric functions. It provides a means to find the angle given the secant value, which is useful in various applications, such as in the analysis of certain types of integrals.

5 Must Know Facts For Your Next Test

1. The inverse secant function, $ extsec^{-1}$, is defined as the angle whose secant is a given value, i.e., $ extsec^{-1}(x) = heta$ such that $ extsec( heta) = x$.
2. The domain of $ extsec^{-1}(x)$ is $x extgeq 1$, as the secant function is only defined for angles in the first and fourth quadrants.
3. The range of $ extsec^{-1}(x)$ is $0 extleq heta extleq rac{ extpi}{2}$, as the secant function is positive in these angles.
4. The inverse secant function is often used to evaluate integrals of the form $ extint rac{dx}{a^2 + x^2}$, which results in $ extsec^{-1}(rac{x}{a})$ plus a constant of integration.
5. The derivative of $ extsec^{-1}(x)$ is $rac{1}{x extsqrt{x^2 - 1}}$, which is useful in differentiating expressions involving the inverse secant function.

Review Questions

• Explain the relationship between the secant function and the inverse secant function, $ extsec^{-1}(x)$.
  • The inverse secant function, $ extsec^{-1}(x)$, is the angle whose secant is the given value $x$. In other words, $ extsec^{-1}(x) = heta$ if and only if $ extsec( heta) = x$. This inverse function allows us to find the angle given the secant value, which is useful in various applications, such as evaluating certain types of integrals involving the secant function.
• Describe the domain and range of the inverse secant function, $ extsec^{-1}(x)$.
  • The domain of the inverse secant function, $ extsec^{-1}(x)$, is $x extgeq 1$, as the secant function is only defined for angles in the first and fourth quadrants. The range of $ extsec^{-1}(x)$ is $0 extleq heta extleq rac{ extpi}{2}$, as the secant function is positive in these angles. This means that the inverse secant function can only take on values within this range, which is an important consideration when working with integrals involving $ extsec^{-1}(x)$.
• Explain how the inverse secant function, $ extsec^{-1}(x)$, is used in the context of integrals resulting in inverse trigonometric functions.
  • In the context of integrals resulting in inverse trigonometric functions, the inverse secant function, $ extsec^{-1}(x)$, is particularly useful in evaluating integrals of the form $ extint rac{dx}{a^2 + x^2}$. By recognizing this integral form, we can apply the substitution $x = a an( heta)$, which leads to $ extint rac{dx}{a^2 + x^2} = a extsec^{-1}(rac{x}{a}) + C$. This integration technique, involving the inverse secant function, is a key skill in solving certain types of integrals encountered in calculus.

Related terms

Inverse Trigonometric Functions:

Inverse trigonometric functions, such as $ extsin^{-1}$, $ extcos^{-1}$, $ exttan^{-1}$, and $ extsec^{-1}$, are used to find the angle given the value of the corresponding trigonometric function.

Secant Function: The secant function, denoted as $ extsec(x)$, is the reciprocal of the cosine function, $ extsec(x) = rac{1}{ extcos(x)}$. It represents the ratio of the hypotenuse to the adjacent side of a right triangle.

Integrals Involving Inverse Trigonometric Functions: Integrals that result in inverse trigonometric functions, such as $ extint rac{dx}{a^2 + x^2}$ which leads to $ extsec^{-1}(rac{x}{a})$, are a specific type of integral encountered in calculus.