Problem 414 - Calculus Volume 1

Question

For the following exercises, find the antiderivatives for the functions. $$\int \frac{d x}{\sqrt{x^{2}+1}}$$

Step-by-Step Solution

Answer

The antiderivative is \( \sinh^{-1}(x) + C \).

Step 1: Identify the Function Type

The given integral is of the form \( \int \frac{dx}{\sqrt{x^2 + 1}} \). Recognize that this is a standard integral that involves the inverse hyperbolic function \( \sinh^{-1}(x) \).

Step 2: Apply Inverse Hyperbolic Sine Rule

Recall the formula for the antiderivative of \( \frac{1}{\sqrt{x^2 + 1}} \), which is \( \sinh^{-1}(x) + C \), where \( C \) is the constant of integration. This stems from the fact that the derivative of \( \sinh^{-1}(x) \) is \( \frac{1}{\sqrt{x^2 + 1}} \).

Step 3: Write the Solution

Directly substitute the known result into the integral. Thus, the antiderivative is \( \int \frac{dx}{\sqrt{x^2 + 1}} = \sinh^{-1}(x) + C \).

Key Concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions are analogues of inverse trigonometric functions. They are particularly useful in solving integrals that involve expressions like \( \sqrt{x^2 + 1} \). Just like inverse trigonometric functions, inverse hyperbolic functions have definitions based on certain algebraic and exponential expressions. For example, the inverse hyperbolic sine function, denoted as \( \sinh^{-1}(x) \), can be defined using a logarithm: \[ \sinh^{-1}(x) = \ln(x + \sqrt{x^2+1}) \]This function is pivotal in calculus particularly because its derivative is \( \frac{1}{\sqrt{x^2 + 1}} \). This property allows us to identify the integral of such a form as a standard antiderivative involving the inverse hyperbolic sine function. Recognizing these patterns helps in solving integrals more efficiently. Knowing when to apply the inverse hyperbolic function transforms not only saves time but also simplifies complex calculations.

Standard Integrals

Standard integrals refer to a set of basic, well-known antiderivatives that appear frequently in calculus. These are essentially predefined solutions to specific types of integral problems, allowing us to quickly apply known results. One of these includes the integral of the form \( \int \frac{dx}{\sqrt{x^2+1}} \).For this particular integral, it is related to the inverse hyperbolic sine function as it follows the derivative formula \( \frac{d}{dx} \sinh^{-1}(x) = \frac{1}{\sqrt{x^2 + 1}} \). Thus, recognizing this pattern allows for the efficient integration of such expressions.Having a good grasp of these standard integrals is crucial, as they act like shortcuts, saving time and reducing errors in calculations. This familiarity also strengthens problem-solving abilities by allowing for quick recognition of integration scenarios.

Constant of Integration

When evaluating indefinite integrals, like the antiderivative \( \int \frac{dx}{\sqrt{x^2 + 1}} \), we always include a constant of integration, denoted as \( C \). This constant represents an entire family of antiderivatives. In calculus, the constant of integration accounts for the fact that differentiation eliminates constants. Since the process of integration is essentially differentiation in reverse, we must add an arbitrary constant to express all possible functions that could differentiate to the given function.Including \( C \) ensures that no solutions are overlooked, embodying the infinite possibilities that the integration process might resolve into. In practical terms, the constant is significant especially in applied mathematics where initial conditions or boundary values are used to find a particular solution among this family.