Problem 42 - University Calculus: Early Transcendentals

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{4+\sqrt{t}}{t^{3}} d t$$

Step-by-Step Solution

Answer

The antiderivative is \(-2t^{-2} - \frac{2}{3}t^{-\frac{3}{2}} + C\).

Step 1: Break down the integral

The integral given is \( \int \frac{4 + \sqrt{t}}{t^3} \, dt \). We can split it into two separate integrals: \( \int \frac{4}{t^3} \, dt + \int \frac{\sqrt{t}}{t^3} \, dt \). This will make it easier to solve each part separately.

Step 2: Simplify the integrals

First, rewrite \( \int \frac{4}{t^3} \, dt \) as \( \int 4t^{-3} \, dt \) and \( \int \frac{\sqrt{t}}{t^3} \, dt \) as \( \int t^{-\frac{3}{2} - 3} \, dt = \int t^{-\frac{5}{2}} \, dt \). This simplifies the expression to power functions which can be integrated using the power rule.

Step 3: Integrate using the power rule

For the integral \( \int 4t^{-3} \, dt \), use the power rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Here, \( n = -3 \), so \( \int 4t^{-3} \, dt = 4 \cdot \frac{t^{-2}}{-2} = -2t^{-2} \). Similarly, for \( \int t^{-\frac{5}{2}} \, dt \), we integrate to find \( \frac{t^{-\frac{3}{2}}}{-rac{3}{2}} = -\frac{2}{3}t^{-\frac{3}{2}} \).

Step 4: Combine the antiderivatives

Combine the results from Step 3: The antiderivative is \(-2t^{-2} - \frac{2}{3}t^{-\frac{3}{2}} + C \), where \( C \) is an arbitrary constant.

Step 5: Verify by differentiation

Differentiate \(-2t^{-2} - \frac{2}{3}t^{-\frac{3}{2}} + C \) to check if we get back to the original integrand. The derivative of \(-2t^{-2} \) is \(4t^{-3} \) and the derivative of \(-\frac{2}{3}t^{-\frac{3}{2}} \) is \( t^{-\frac{5}{2}} \), adding up gives \( \frac{4 + \sqrt{t}}{t^3} \), which matches the original integrand.

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus, especially when dealing with differentiation and integration tasks. It provides a simple method to find the derivative or antiderivative of expressions that are power functions.

• Differentiation: When differentiating a power function of the form \( x^n \), the power rule states that the derivative is \( nx^{n-1} \). This rule helps in breaking down and simplifying complex expressions.


• Integration (Antiderivatives): Reversing the differentiation process, the power rule also applies when integrating. For a given power function \( x^n \), the indefinite integral or antiderivative is \( \frac{x^{n+1}}{n+1} + C \), with \( C \) being the constant of integration.

To apply this, ensure that no denominator of zero occurs, namely avoiding \( n eq -1 \). This rule is your go-to tool in integral calculus, facilitating quick solutions for polynomial terms.

Differentiation

Differentiation is one of the core operations in calculus, used to determine how a function changes as its input changes. It involves calculating the derivative, which represents an instantaneous rate of change.

• Purpose of Differentiation: Find the rate of change or slope of a function at any given point. It's widely used in various fields such as physics for motion analysis, economics for cost functions, and biology for population dynamics.


• Common Rules: Rules like the power rule, product rule, and chain rule simplify the differentiation process for different types of functions. In the solution, verifying the antiderivative by differentiation ensures the correctness of the integration process.

For our function, differentiating the expression \(-2t^{-2} - \frac{2}{3}t^{-\frac{3}{2}} + C\), reverts it back to the original form \(\frac{4+\sqrt{t}}{t^3}\), verifying that our process of finding the antiderivative was indeed correct.

Indefinite Integral

The indefinite integral is an essential concept when learning about antiderivatives. It represents a family of functions that differ by a constant, providing solutions to integration problems without specific bounds.

• Antiderivative: It involves reversing the differentiation process, essentially finding the original function given its derivative. This results in a function plus an arbitrary constant \( C \).


• Notation: Denoted by the integral sign \( \int \), it implies summation over an indefinite range. The result is not unique due to the constant \( C \), representing all possible functions within that family.

For example, in our exercise, the integral \(\int \frac{4+\sqrt{t}}{t^3} \, dt\) is broken down and solved using the power rule, resulting in the indefinite integral: \(-2t^{-2} - \frac{2}{3}t^{-\frac{3}{2}} + C\). The process, broken into detailed steps, allows students to see the flexibility and generality of solutions using indefinite integrals.