Problem 102 - Calculus for Biology and Medicine

Question

Evaluate the definite integrals. $$ \int_{4}^{9} \frac{1+\sqrt{x}}{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The value of the definite integral is 7.

Step 1: Simplify the integrand

Observe that the integrand \( \frac{1+\sqrt{x}}{\sqrt{x}} \) can be simplified. Write this expression as two separate fractions: \( \frac{1}{\sqrt{x}} + \frac{\sqrt{x}}{\sqrt{x}} \). This simplifies further to \( x^{-1/2} + 1 \).

Step 2: Integrate the simplified expression

Now, integrate each term separately. Use the rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).- For \( x^{-1/2} \): \[ \int x^{-1/2} \, dx = \frac{x^{1/2}}{1/2} = 2\sqrt{x} \]- For the constant 1: \[ \int 1 \, dx = x \]

Step 3: Evaluate the antiderivative at the bounds

Combine the antiderivatives:\[ F(x) = 2\sqrt{x} + x \]Evaluate this antiderivative from \( x = 4 \) to \( x = 9 \) using the Fundamental Theorem of Calculus:\[ F(9) = 2\sqrt{9} + 9 = 6 + 9 = 15 \]\[ F(4) = 2\sqrt{4} + 4 = 4 + 4 = 8 \]

Step 4: Compute the definite integral

Apply the Fundamental Theorem of Calculus, which states that the value of the definite integral is \( F(9) - F(4) \):\[ F(9) - F(4) = 15 - 8 = 7 \]

Key Concepts

Understanding Antiderivatives

One of the building blocks in calculus is the concept of an antiderivative. Simply put, an antiderivative is a function whose derivative equals the given function. When we perform the integration, we're essentially looking for such a function. Imagine antiderivatives as the opposite action of differentiation. If differentiation gives you the slope of a curve at any point, finding the antiderivative helps you reconstruct the curve from its slope pattern.
For example, when we integrate the simplified expression in our problem, such as \(x^{-1/2} + 1\), we are finding its antiderivative. Remember, rather than focusing on finding a singular solution, we often refer to "family" of functions with a constant denoted as \(C\) because several curves can share the same slope but differ by a vertical shift.

• The antiderivative of \(x^{-1/2}\) is \(2\sqrt{x}\).
• The antiderivative of the constant 1 is \(x\).

This process reveals a new view of functions beyond mere differentiation.

Applying Integration Rules

Integration involves various rules to simplify the calculation of antiderivatives. Integration rules are essentially the cheat sheet for handling different types of terms within a function. One of the most widely used is the power rule for integration. This rule is especially handy in this problem where we're dealing with powers of \(x\). The rule states:
\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n\) is not equal to -1.
This is vital because it gives us a formula to follow which tells us how to systematically "undo" the differentiation of powers of \(x\).

• For instance, when integrating \(x^{-1/2}\), the power increases by 1 (turning to \(x^{1/2}\)) and gets divided by the new power (1/2). Hence, the integral becomes \(2\sqrt{x}\).
• For a constant such as 1, remember that its integration simply returns "x".

Integration rules like these transform daunting integrals into solvable steps.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is an incredible bridge connecting differentiation and integration. It tells us that differentiation and integration are inverse processes. This theorem is not just theoretical; it's extremely practical for evaluating definite integrals. Essentially, it provides a method to evaluate the total accumulation, like the area under a curve, by finding the antiderivative.
In problems involving definite integrals, like ours, the theorem provides a streamlined way to solve them. It comprises two parts:

• The first part establishes that if \(F\) is the antiderivative of \(f\), then the succession of integrating \(f\) and then differentiating \(F\) gets back to \(f\)
• The second part involves using the antiderivative to compute the definite integral from \(a\) to \(b\), i.e., \(F(b) - F(a)\).

For our example, once the antiderivatives are combined, \(F(x) = 2\sqrt{x} + x\). Simply calculate \(F(9)\) and \(F(4)\), and subtract to find the integral's value: 7. This seamless calculation using the theorem makes evaluating definite integrals straightforward.