Problem 23 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-26\) $$ \int_{1}^{\sqrt{2}} \frac{s^{2}+\sqrt{s}}{s^{2}} d s $$

Step-by-Step Solution

Answer

The value of the integral is 1.

Step 1: Simplify the Integrand

Start by simplifying the integrand. The given function is \( \frac{s^2 + \sqrt{s}}{s^2} \). Split it into two separate terms: \( \frac{s^2}{s^2} + \frac{\sqrt{s}}{s^2} \). This simplifies to \( 1 + \frac{1}{s^{3/2}} \).

Step 2: Integrate Each Term Separately

Now, integrate the expression term by term. The integral becomes \( \int_1^{\sqrt{2}} \left( 1 + \frac{1}{s^{3/2}} \right) \, ds \). This can be written as \( \int_1^{\sqrt{2}} 1 \, ds + \int_1^{\sqrt{2}} \frac{1}{s^{3/2}} \, ds \).

Step 3: Compute the First Integral

The first integral is \( \int_1^{\sqrt{2}} 1 \, ds = s \Big|_1^{\sqrt{2}} = \sqrt{2} - 1 \).

Step 4: Compute the Second Integral

For the second integral, \( \int_1^{\sqrt{2}} \frac{1}{s^{3/2}} \, ds \), rewrite the integrand as \( s^{-3/2} \). The antiderivative of \( s^{-3/2} \) is \( \frac{s^{-1/2}}{-1/2} = -2s^{-1/2} \). Compute the definite integral: \(-2s^{-1/2} \Big|_1^{\sqrt{2}} = -2(\frac{1}{\sqrt{2}}) + 2(1) = -\sqrt{2} + 2 \).

Step 5: Combine the Results

Add the results from Step 3 and Step 4: \( (\sqrt{2} - 1) + (-\sqrt{2} + 2) = 1 \).

Key Concepts

Simplifying Integrals

Before diving into more complex integration, it's essential to simplify the integrand, or the function inside the integral. Simplifying helps make the integration process easier and reduces mistakes.
To simplify, you might need to separate or recombine terms. In the given exercise, we started with the expression \( \frac{s^2 + \sqrt{s}}{s^2} \).
To simplify, we split the expression into two distinct terms:

• \( \frac{s^2}{s^2} \), which simplifies to \( 1 \)
• \( \frac{\sqrt{s}}{s^2} = \frac{1}{s^{3/2}} \).

This step was critical to breaking down the problem and preparing it for easier integration. Often, simplifying involves looking for opportunities to cancel terms or rewrite expressions to more manageable formats.

Integration Techniques

Integration techniques are strategies to solve integrals that aren't straightforward to compute. Simple sums of functions can often be split apart to make the process easier.
In our exercise, the simplified integrand became \(1 + \frac{1}{s^{3/2}}\). From there, integrating each term separately was a strategy used.
For this purpose:

• The first integral is \(\int 1 \, ds\), which is straightforward since the antiderivative of \(1\) is \(s\).
• The second part, \(\int \frac{1}{s^{3/2}} \, ds\), involves recognizing the expression \(\frac{1}{s^{3/2}}\) can be rewritten as \(s^{-3/2}\).

These terms helped in systematically applying different basic integration techniques. Mastering these techniques is critical in tackling more complex forms of integrals.

Antiderivatives

The concept of an antiderivative is at the heart of integration. An antiderivative of a function is a function whose derivative is the original function we started with. For definite integrals, we use antiderivatives to evaluate the area under the curve between specified bounds.
In this exercise, we needed the antiderivatives for each simplified term:

• For \(1\), the antiderivative is simply \(s\).
• For \(s^{-3/2}\), the antiderivative is obtained by reversing the power rule of derivatives: \[\int s^{-3/2} \, ds = \frac{s^{-1/2}}{-1/2} = -2s^{-1/2}\].

It's crucial to remember that once the antiderivatives are known, we compute the definite integral by evaluating the antiderivative at the upper limit and subtracting the evaluation at the lower limit. This process results in the definite integral value, giving us a useful result in applications like calculating areas or solving physics problems.