Question

Use tables to perform the integration. $$ \int \frac{d x}{\sqrt{4 x+1}} $$

Short answer

The integral is \( \sqrt{4x + 1} + C \).

Step-by-step solution

Identify the Integral Form

The given integral is \( \int \frac{d x}{\sqrt{4 x+1}} \). By comparing this with standard integral forms, it resembles the form \( \int \frac{1}{\sqrt{ax+b}} \, dx \). From this, we identify that \( a = 4 \) and \( b = 1 \).

Use Integral Tables

The integral \( \int \frac{1}{\sqrt{ax+b}} \, dx \) has a known result, which can be found in integral tables. According to such tables, the integral is \( \frac{1}{\sqrt{a}} \times 2\sqrt{ax + b} + C \).

Substitute Values

Now, substitute \( a = 4 \) and \( b = 1 \) into the result from the integral table: \[ \frac{1}{\sqrt{4}} \times 2\sqrt{4x + 1} + C = \frac{1}{2} \times 2\sqrt{4x + 1} + C \].

Simplify the Expression

Simplify the expression obtained: \( \frac{1}{2} \times 2\sqrt{4x + 1} + C = \sqrt{4x + 1} + C \). This is the solution to the integral.

Key concepts

Definite Integrals

Definite integrals are used to calculate the area under a curve between two specific points on the x-axis. They are written in the form \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration. Unlike indefinite integrals, definite integrals result in a numerical value rather than a function. The process involves evaluating the antiderivative of the function at the boundaries and then subtracting these values. This can be visualized as finding the total accumulation of quantities, such as distance when given velocity.

• The fundamental theorem of calculus connects differentiation and integration, providing a method to evaluate definite integrals.
• Definite integrals are useful in physics and engineering for calculating work, electrical charge, and more.

When dealing with definite integrals, always pay attention to the limits and ensure that they are correctly applied after finding the antiderivative.

Indefinite Integrals

Indefinite integrals are the reverse process of differentiation, aiding in finding the original function given its derivative. It is expressed as \( \int f(x) \, dx \), which results in a family of functions differing by a constant, noted as \( C \). This represents the most general form of the original function. To solve an indefinite integral, integration techniques such as power rule, substitution, and parts may be used.

• Indefinite integrals do not have set limits, which means the result is a function plus a constant \( C \), representing any constant value that could have been differentiated away.
• They're essential for solving differential equations and in physics for evaluating motion equations and potential functions.

Remember, finding the indefinite integral is akin to "undoing" differentiation.

Integral Tables

Integral tables serve as a reference to quickly find the antiderivative of common functions, which is incredibly helpful when solving complex integrals. These tables list integrals of standard forms and can simplify the process of integration. For example, the table provides the integral \( \int \frac{1}{\sqrt{ax+b}} \, dx \) as \( \frac{1}{\sqrt{a}} \times 2 \sqrt{ax + b} + C \).

• They eliminate the need to derive integrals from scratch every time, saving valuable time and effort.
• For uncommon integrals, breaking down the function using algebraic manipulation or substitution before consulting the table may be necessary.

Using integral tables effectively requires recognizing the form of the function being integrated and matching it with the appropriate entry in the table.

Substitution Method

The substitution method is a technique used when integrating functions, particularly effective for dealing with integrals involving composite functions. The goal is to simplify the integral by substituting part of it with a single variable, often denoted as \( u \). This transformation can make the integral easier to solve.

• Choose a substitution, \( u = g(x) \), that simplifies part of the integrand, typically the inner function of a composition.
• Change the differential: if \( u = g(x) \), then \( du = g'(x) \, dx \).
• Reformulate the integral in terms of \( u \), solve it, and substitute back \( g(x) \) before finalizing.

Substitution is particularly useful when dealing with expressions inside a radical or a rational function, simplifying the integral to a form that is more straightforward to solve.