Problem 22 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 21 through 30 , evaluate the indicated definite integral. \(\int_{1}^{4}\left(\sqrt{t}+t^{-3 / 2}\right) d t\)

Step-by-Step Solution

Answer

Evaluate using the same range as shown. Combine to return the same value usually integrating vertically

Step 1 - Split the integral

First, split the given integral into two separate integrals: \[ \begin{aligned} \ \ \ \ \ &\int_{1}^{4}\bigg(\root t\bigg)+t^{-3 / 2} dt \ & = \int_{1}^{4} \root dt +\bigg( \int_{1}^{4} t^{-3 / 2} \ & dt \bigg) \end{aligned} \] Each of these will be solved separately. Start with the first integral \(\int_{1}^{4} \root dt \)

Step 2 - Solve the first integral

Conveniently break the integral into different constituents for integration.

Step 3 - Solve the simplified integral, converting into red

\[ \int_{1}^{4} t^{1/2} dt \] using the power rule \( \int t^n dt = \frac{t^{n+1}}{n+1} \)

Step 4 - Apply power rule for integration

Evaluate using integral of root step by step in integration.

Step 5 - Evaluate the first integral

evaluating as indcatedo scientifically

Step 6 - evaluating the integration

Using 1 to 4 range and check the root power evaluation to approximate the retuns

Step 7 - use power rule for 2nd integration

Apply the same power rule completing the same integral step by step : \[ evaluate integration range from 1 -4 \]

Step 8 - Compute the final range evaluation

same rule at different step check evaluating through return integration evaluating using limits

Step Final : Compute resulting value

Compute complete power and return as final answer usually integral value check:

Key Concepts

integral splitting

To start evaluating the definite integral \( \int_{1}^{4}( \sqrt{t} + t^{-3/2})dt \ \), we use the technique known as integral splitting. This technique involves breaking down a complex integral into simpler, more manageable parts. In this case, we can split the given integral into two separate integrals: \[ \int_{1}^{4} \sqrt{t} \ dt + \int_{1}^{4} t^{-3/2} \ dt \ \] By splitting the integral, we simplify our task. Each of these integrals can now be evaluated individually.
This method is quite useful for integrating composite functions.

power rule in integration

Once we have split the integral, we can use the power rule of integration for each part. The power rule states if \ n \ does not equal \ -1 \, then: \[ \int t^n \ dt = \frac{t^{n+1}}{n+1} + C \ \] For our first split integral, let's rewrite \ \sqrt{t} \ as \ t^{1/2}: \[\int_{1}^{4} t^{1/2} \ dt \ \] Applying the power rule, we find: \[ \int t^{1/2} \ dt = \frac{t^{(1/2)+1}}{(1/2)+1} + C = \frac{t^{3/2}}{3/2} = \frac{2}{3} t^{3/2} \ \] For the second split integral, we have: \[ \int_{1}^{4} t^{-3/2} dt \ \] Applying the power rule again: \[ \int t^{-3/2} \ dt = \frac{t^{(-3/2)+1}}{(-3/2)+1} + C = \frac{t^{-1/2}}{-1/2} = -2 t^{-1/2} = -2 \frac{1}{ \sqrt{t}} \ \]

definite integrals

Now that we have our antiderivatives for each part, let's evaluate them over the definite integral’s limits from 1 to 4. For the first split integral: \[ \frac{2}{3}[t^{3/2}]_{1}^{4} \ = \frac{2}{3}(4^{3/2}) - \frac{2}{3}(1^{3/2}) \ \] Simplify the expression:

• \frac{2}{3} \cdot 8 - \frac{2}{3} \cdot 1 = \frac{16}{3} - \frac{2}{3} = \frac{14}{3}

For the second split integral: \[ -2[t^{-1/2}]_{1}^{4} = -2 \left( \frac{1}{ \sqrt{4}} - \frac{1}{ \sqrt{1}} \right) \ = -2(\frac{1}{2} - 1) \ \] Simplify the expression:

• -2(- \frac{1}{2}) = 1 \

Finally, we sum up the results from both integrals:

• \frac{14}{3} + 1 = \frac{14}{3} + \frac{3}{3} = \frac{17}{3}

Thus, the value of the definite integral \ \int_{1}^{4}(\sqrt{t} + t^{-3/2})dt \ \ is \ \frac{17}{3} \.