Question

Use a table of integrals to evaluate the following integrals. $$ \int_{0}^{4} \frac{x}{\sqrt{1+2 x}} d x $$

Short answer

The value of the integral is \( \frac{10}{3} \).

Step-by-step solution

Identify the Integral Form

The given integral is \( \int \frac{x}{\sqrt{1+2x}} \, dx \). Look for an entry in the table of integrals that resembles this form. A suitable form related to \( \int \frac{x}{\sqrt{ax+b}} \, dx \) could be useful.

Use Substitution to Match a Known Integral

To match a standard form, consider a substitution that simplifies the expression inside the integral. Let \( u = 1 + 2x \), so \( du = 2 \, dx \). Hence, \( dx = \frac{1}{2} \, du \). You also have \( x = \frac{u - 1}{2} \). Change the limits of integration: when \( x = 0, \ u = 1 \) and when \( x = 4, \ u = 9 \). Substitute into the integral to get: \[ \int_{1}^{9} \frac{(u-1)/2}{\sqrt{u}} \cdot \frac{1}{2} \, du \].

Simplify the Integral

Simplify the integral: \[ \frac{1}{4} \int_{1}^{9} \frac{u-1}{\sqrt{u}} \, du = \frac{1}{4} \int_{1}^{9} \left( \sqrt{u} - \frac{1}{\sqrt{u}} \right) \, du \]. This breaks into two separate integrals: \[ \frac{1}{4} \left( \int_{1}^{9} u^{1/2} \, du - \int_{1}^{9} u^{-1/2} \, du \right) \].

Apply Standard Integral Formulas

Use standard integral formulas: \[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \]. So, \( \int u^{1/2} \, du = \frac{2}{3} u^{3/2} \) and \( \int u^{-1/2} \, du = 2u^{1/2} \). Applying these, we have: \[ \frac{1}{4} \left( \frac{2}{3} u^{3/2} \bigg|_1^9 - 2u^{1/2} \bigg|_1^9 \right) \].

Evaluate the Definite Integral

Evaluate each part at the bounds: \[ \frac{1}{4} \left( \frac{2}{3} (9^{3/2} - 1^{3/2}) - 2(9^{1/2} - 1^{1/2}) \right) \]. Calculate: \( 9^{3/2} = 27 \) and \( 1^{3/2} = 1 \); \( 9^{1/2} = 3 \) and \( 1^{1/2} = 1 \). This results in: \[ \frac{1}{4} \left( \frac{2}{3} (26) - 2(2) \right) = \frac{1}{4} \left( \frac{52}{3} - 4 \right) = \frac{1}{4} \left( \frac{52}{3} - \frac{12}{3} \right) = \frac{1}{4} \left( \frac{40}{3} \right) = \frac{10}{3} \].

Key concepts

Definite Integral

In calculus, a definite integral differs from an indefinite integral because it evaluates the area under a curve between two specific points, giving an actual numerical value rather than a family of functions. The definite integral of a function \( f(x) \) from \( a \) to \( b \) is denoted as \( \int_{a}^{b} f(x) \, dx \). This notation means we are summing up all the infinitesimal areas under the curve between \( x = a \) and \( x = b \).
To solve a definite integral, we typically perform these steps:

• Find the antiderivative (indefinite integral) of the function.
• Evaluate at both limits of integration and subtract the values: \( F(b) - F(a) \).

One key point to understand is that definite integrals can represent various real-world quantities such as distances, areas, and even probabilities, depending on the context of the problem.

Substitution Method

The substitution method is a frequently used technique in calculus to simplify an integral by introducing a new variable. This method is particularly beneficial if the integral contains a composition of functions that makes it complex. Here’s how you can tackle such integrals using substitution:

• Identify a portion of the integral that can be replaced with a single variable \( u \). This is often the part inside a composite function or nested inside a square root or exponent.
• Determine the derivative \( du \) of the substitution and express \( dx \) in terms of \( du \).
• Change the limits of integration if working with a definite integral, using the substitution equation.
• Substitute \( u \) and \( du \) into the integral, simplifying it to a form that is easier to integrate.

By transforming the integral into this simpler form, you can often use known integral formulas to solve. In our example, we used \( u = 1 + 2x \) to simplify the expression and allow for a straightforward integration.

Table of Integrals

A table of integrals is like a treasure map for calculus students. It provides a list of frequently encountered integrals and their antiderivatives. Instead of always calculating integrals from scratch, you can look up similar integral forms and use them directly, saving time and errors. Here’s how to effectively use a table of integrals:

• Identify the form of your integral and find a corresponding form in the table. Many tables categorize integrals by types of functions, such as trigonometric, exponential, or algebraic.
• If your integral does not match exactly, consider using algebraic manipulations or a substitution method to transform it into a standard form.
• Apply the corresponding result from the table to find the antiderivative directly.

Using a table of integrals is especially helpful in a time-constrained setting, like an exam or complex problem-solving, where speed and accuracy save valuable resources. In our exercise, the use of the table was crucial to identify the form \( \int \frac{x}{\sqrt{ax+b}} \, dx \) and solve it efficiently.