Question

For the following exercises, find the definite or indefinite integral. $$ \int_{0}^{1} \frac{d x}{3+x} $$

Short answer

The definite integral is \( \ln\left( \frac{4}{3} \right) \).

Step-by-step solution

Identify the Type of Integral

This is a definite integral as we are computing the integral of the function from 0 to 1. It's defined with limits on the integral.

Recall the Integration Formula for Logarithm

The integral of \( \frac{1}{x+a} \) is \( \ln|x+a| + C \), where \( C \) is the constant of integration. In this case, it simplifies since we are dealing with definite integration.

Apply the Integration Formula

Apply the formula to integrate \( \frac{1}{3+x} \). We obtain \( \int \frac{d x}{3 + x} = \ln|3 + x| \).

Evaluate the Definite Integral

Evaluate the integral from 0 to 1: \[\int_0^1 \frac{d x}{3 + x} = \left[ \ln|3 + x| \right]_0^1.\]This gives us: \[\ln|3+1| - \ln|3+0| = \ln(4) - \ln(3).\]

Simplify the Result

Use logarithmic properties to simplify:\[\ln(4) - \ln(3) = \ln\left( \frac{4}{3} \right).\]

Final Answer

The final answer for the definite integral is \( \ln\left( \frac{4}{3} \right) \).

Key concepts

Definite Integrals

Definite integrals are a way to calculate the accumulation of a quantity, like area under a curve, over a specific interval. In our exercise, we looked at the function \( \frac{1}{3+x} \) from \( 0 \) to \( 1 \). This specific interval is crucial because it defines the limits within which we calculate the integral.

The result of a definite integral is a number, which represents the total accumulation between the given limits. It’s different from indefinite integrals, which yield a function plus a constant. In our exercise, the definite integral ultimately gave us a specific numerical result: \( \ln\left( \frac{4}{3} \right) \).

Key points to remember about definite integrals:

• Have specific upper and lower limits, represented as numbers at the top and bottom of the integral sign.
• Provide a numerical output representing accumulated quantity.
• Can be evaluated using fundamental properties of integral calculus, such as the Fundamental Theorem of Calculus.

Logarithmic Integration

Logarithmic integration is used when dealing with integrals involving expressions that simplify into a natural logarithm. The integral \( \frac{1}{3+x} \) is an example, where recognizing the form \( \frac{1}{x+a} \) helps us apply the logarithmic formula.

The formula \( \int \frac{1}{x+a} \, dx = \ln|x+a| + C \) is of particular importance. When using definite integrals, the constant \( C \) can be omitted because the evaluation is over a specific interval. This makes logarithmic integration a powerful and convenient tool for definite integrals.

Tips for effective logarithmic integration:

• Look for the \( \frac{1}{x+a} \) structure, which reveals where the logarithmic rule applies.
• Ensure proper handling of absolute values, particularly important in definite integrals where limits might change the sign.
• Remember that definite integrals over symmetrical limits can sometimes simplify calculation.

Mathematical Steps in Integration

Mastering integration involves following key mathematical steps to transition from a function to its integral, either definite or indefinite. Here's how it worked in this exercise step-by-step:

• Identifying the Integral Type: Recognize whether you're dealing with a definite or indefinite integral. Our case was definite, signified by the limits \( 0 \) to \( 1 \).
• Applying the Correct Formula: Knowing when to use integration formulas like logarithms is critical. For \( \frac{1}{3+x} \), we used the logarithmic form \( \ln|3+x| \).
• Evaluating the Expression: Substitute the upper and lower limits \( 1 \) and \( 0 \) respectively, into the integrated formula. Thus, \( \left[ \ln|3+x| \right]_0^1 \).
• Simplifying the Result: Using logarithmic properties, simplify \( \ln(4) - \ln(3) \) to \( \ln\left( \frac{4}{3} \right) \).

Understanding each of these steps helps in accurately solving integrals and ensuring mathematical processes are handled correctly.