Question

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

Short answer

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

Step-by-step solution

Identify the Integral

We need to find the antiderivative of the function \( \int \frac{1}{\sqrt{x^2+1}} \, dx \). This is an integral that resembles a standard form.

Recognize the Standard Form

The integral \( \int \frac{1}{\sqrt{x^2+1}} \, dx \) is known to be a standard form. The antiderivative of this specific function is related to the logarithmic or inverse hyperbolic functions.

Apply the Solution Formula

According to standard calculus results, the antiderivative of \( \int \frac{1}{\sqrt{x^2+1}} \, dx \) is given by \( \sinh^{-1}(x) + C \), where \( C \) is the constant of integration.

Write the Final Answer

From the previous step, we have computed that the antiderivative is \( \sinh^{-1}(x) + C \). Therefore, the solution to the integral is complete.

Key concepts

Integral Calculus

Integral calculus is a fundamental part of calculus that focuses on the accumulation of quantities. It involves finding antiderivatives, also known as indefinite integrals. While differentiation is about finding instantaneous rates of change, integration is about finding total quantities. When you integrate a function, you are essentially reversing the process of differentiation.

There are two main types of integrals: definite and indefinite. While definite integrals calculate the area under the curve between specific limits, indefinite integrals (as in our example) find the general form or family of antiderivatives without specific bounds. For instance, if you have the function \( \frac{1}{\sqrt{x^2+1}} \), you're finding which function, when differentiated, gives this result.

• Basic principle: Finding the original function before it was differentiated.
• Standard forms: Many integrals have known results, making complex calculations simpler, as seen with \( \sinh^{-1}(x) \).
• Integration techniques: Techniques like substitution, integration by parts, and recognizing standard forms are crucial components.

Inverse Hyperbolic Functions

Inverse hyperbolic functions are akin to inverse trigonometric functions, but they deal with hyperbolic functions such as sinh, cosh, and tanh. These functions have significant applications in calculus, especially in integral calculus where they appear frequently in integration problems.

Hyperbolic functions resemble trigonometric functions but are based on hyperbolas, not circles. The inverse hyperbolic sine, \( \sinh^{-1}(x) \), which appears in our example, gives the value whose hyperbolic sine is \( x \). This function is particularly useful in integration, as it often emerges in integrals involving quadratic expressions and square roots, like \( \sqrt{x^2 + 1} \).

• Definition: Inverse of the hyperbolic function "sinh." Represents arcsinh \( x \).
• Formula: \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2+1}) \).
• Applications: Solving integrals, physics problems, and engineering tasks involving exponential growth.

Constant of Integration

The constant of integration, denoted as \( C \), is a crucial element in indefinite integrals. It represents the set of all possible antiderivatives. Since differentiation of a constant gives zero, when finding antiderivatives, it's essential to account for this possible constant.

Whenever you compute an indefinite integral, you're actually identifying a family of functions. Each member of this family differs from others by a constant value. This constant can be any real number, which makes the solution flexible until further information (like initial conditions) is provided.

• Indefinite integrals: Always include \( C \) when identifying antiderivatives.
• Definite vs. indefinite: Only indefinite integrals have this constant; definite integrals use specific boundaries instead.
• Importance: Represents infinite sets of solutions tailored to specific problems when conditions are applied.