Problem 15 - Calculus: Early Transcendental Functions

Question

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{1}^{4} \frac{x-3}{x} d x$$

Step-by-Step Solution

Answer

The exact value of the integral is \( 3 - 3ln4 \)

Step 1: Simplify the integrand

The integrand \(\frac{x-3}{x}\) can be simplified by dividing each term by x. This makes the integral easier to solve: \( \int_{1}^{4} (\frac{x}{x}-\frac{3}{x}) dx = \int_{1}^{4} (1-\frac{3}{x}) dx\)

Step 2: Find the antiderivative of the simplified integrand

Separate the integral into two: \( \int_{1}^{4} 1 dx - \int_{1}^{4} \frac{3}{x} dx \). The antiderivatives are: \( F(x) = x \) for \( \int 1 dx \) and \( F(x) = -3ln|x| \) for \( \int \frac{3}{x} dx \)

Step 3: Apply the Fundamental Theorem of Calculus

Substitute the values into the Fundamental Theorem of Calculus: \((x - 3ln|x|)\Big|_1^4 \). When you substitute the bounds of the integral, you should get: \( (4 - 3ln4) - (1 - 3ln1) = 3 - 3ln4 \)

Key Concepts

Definite Integral

The definite integral is a fundamental concept in calculus that represents the accumulation of quantities and can be considered as the signed area under a curve. When we look at an expression such as \(\int_{a}^{b} f(x) \, dx\), it represents the definite integral of the function \(f(x)\) within the interval [a, b]. To evaluate this, we calculate the antiderivative (also known as the indefinite integral) of \(f(x)\) and then apply the limits of the integral, which is neatly encapsulated in the Fundamental Theorem of Calculus.

In the exercise example, the definite integral from 1 to 4 of \(\frac{x-3}{x}\) dx, represents the area under the curve of \(\frac{x-3}{x}\) from x=1 to x=4. The subtraction of values obtained by plugging the upper limit and lower limit into the antiderivative function provides the value of the definite integral.

Antiderivative

An antiderivative, also known as an indefinite integral, is essentially the inverse operation to differentiation. If F(x) is an antiderivative of f(x), then the derivative of F(x) with respect to x is f(x), which is written mathematically as \(F'(x) = f(x)\).

The antiderivative is used in finding definite integrals as demonstrated by the Fundamental Theorem of Calculus which states that if F is an antiderivative of f on an interval [a, b], then \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\). In the provided exercise, the antiderivatives of the simplified integrands 1 and \(\frac{3}{x}\) have been found as x and \( -3\ln|x|\) respectively. These are utilized in the final step to evaluate the entire definite integral.

Integrand Simplification

Integrand simplification is a technique used in integral calculus to make the process of finding the antiderivative more straightforward. This often involves algebraic manipulation such as dividing through by common factors, expanding expressions, or applying trigonometric identities.

Within the step by step solution provided, the integrand \(\frac{x-3}{x}\) is simplified by dividing each term by x, resulting in the expression 1 - \(\frac{3}{x}\). Simplifying the integrand before integrating makes it possible to readily identify the antiderivatives of the separate terms. This simplification is a crucial step and helps in breaking down more complex problems into more manageable pieces that can be solved with basic integration rules.