Question

Evaluate the following definite integrals. $$ \int_{e}^{e^{2}} \frac{1}{t+3} d t $$

Short answer

Question: Find the value of the definite integral $$\int_{e}^{e^{2}} \frac{1}{t+3} d t$$. Answer: The value of the definite integral is $$\ln \left| \frac{e^{2}+3}{e+3} \right|.$$

Step-by-step solution

Find the antiderivative of the function

To find the antiderivative of the function, we need to perform the integration of the function with respect to t. The given function is a rational function, so in this case, we can see that it is already in the form that can be integrated with a simple logarithm. We will find the antiderivative of the given function: $$ \int \frac{1}{t+3} d t $$ It looks like the natural logarithm function, so let's rewrite and integrate it: $$ \int \frac{1}{t+3} d t = \ln|t+3| + C $$ Where C is the constant of integration.

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if we have a definite integral, we can calculate it by evaluating the antiderivative at the upper bound and subtracting the value of the antiderivative at the lower bound. So, let's apply the Fundamental Theorem of Calculus to our problem: $$ \int_{e}^{e^{2}} \frac{1}{t+3} d t = \left[ \ln|t+3| \right]_{e}^{e^{2}} $$ Now we will plug in our upper and lower bounds: $$ = \ln|e^{2}+3| - \ln|e+3| $$

Simplify the expression

To simplify the expression, we can use the properties of logarithms. In particular, we can use the property that the difference of two logarithms is the logarithm of the quotient of their arguments: $$ \ln|e^{2}+3| - \ln|e+3| = \ln \left| \frac{e^{2}+3}{e+3} \right| $$

Write the final answer

The final result after evaluating the definite integral is: $$ \int_{e}^{e^{2}} \frac{1}{t+3} d t = \ln \left| \frac{e^{2}+3}{e+3} \right| $$

Key concepts

Antiderivative

The concept of an antiderivative is central to the process of integration. An antiderivative, also known as an indefinite integral, is a function that reverses the process of differentiation. If the derivative of function F is f, then F is an antiderivative of f. It's important to note that antiderivatives are not unique; adding a constant to an antiderivative yields another valid antiderivative. This is why we include the constant of integration, usually denoted as C, when finding antiderivatives.

In the given exercise example, to find the antiderivative of the function \(\frac{1}{t+3}\), one must integrate the function with respect to t, resulting in \(\ln|t+3| + C\), where C represents the constant of integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a powerful connection between the processes of differentiation and integration. It consists of two parts. The first part guarantees the existence of antiderivatives for continuous functions, while the second part—more relevant to evaluating definite integrals—states that if a function f is continuous over the interval [a, b] and F is an antiderivative of f, then the definite integral of f from a to b is equal to F(b) - F(a).

In practice, this theorem allows us to evaluate the definite integral by finding the antiderivative and then subtracting the antiderivative evaluated at the lower limit from the antiderivative evaluated at the upper limit. Applying this principle in the exercise, the evaluation goes from complex integral calculus to relatively simple substitution, demonstrating the theorem's pivotal role in simplifying the integral evaluation process.

Logarithmic Integration

Logarithmic integration is a technique used when integrating functions that can be expressed as the inverse of a linear function, commonly taking the form \(\frac{1}{x}\). When we see an integral that resembles the form \(\int \frac{1}{t} dt\), we can think of the natural logarithm function due to the relationship between derivatives of the logarithm and the integral of \(1/x\).

The general integral \(\int \frac{1}{x} dx\) becomes \(\ln|x| + C\), taking advantage of the derivative of ln(x), which is \(1/x\). In the textbook problem, applying logarithmic integration transforms the integral of a simple rational function into the natural logarithm of the absolute value of the variable plus the constant, streamlining the computation significantly. This approach is often used when integrating rational functions that can be adjusted to fit this form.

Rational Function Integration

Rational function integration deals with the integration of rational functions, which are the ratios of polynomials. The integration of rational functions can be more complex when the degree of the numerator is equal to or greater than the degree of the denominator. In such cases, long division or partial fraction decomposition may be required.

However, the integral given in the exercise, \(\int_{e}^{e^{2}} \frac{1}{t+3} dt\), is already in a form that allows for straightforward integration because the degree of the numerator is less than the degree of the denominator. This simplicity means we can integrate directly without additional algebraic manipulation, as shown in the step-by-step solution. Understanding the various methods to integrate rational functions is valuable, as it prepares students to handle more complicated scenarios that they may encounter.