Problem 40 - 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. $$\int x^{-3}(x+1) d x$$

Step-by-Step Solution

Answer

The antiderivative is \(-\frac{1}{x} - \frac{1}{2x^2} + C\).

Step 1: Distribute the Power

First, distribute the term \(x^{-3}\) across \((x+1)\) in the integrand. This simplifies the expression:\[x^{-3}(x+1) = x^{-3} \cdot x + x^{-3} \cdot 1 = x^{-2} + x^{-3}\]

Step 2: Integrate Term by Term

Integrate each term separately using the power rule for integration. The power rule is given by \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), where \(n eq -1\). Applying this rule:\[\int x^{-2} dx = \frac{x^{-1}}{-1} = -\frac{1}{x}\]\[\int x^{-3} dx = \frac{x^{-2}}{-2} = -\frac{1}{2x^2}\]

Step 3: Combine Integrals and Add Constant of Integration

Combine the results from the previous step and add the constant of integration \(C\):\[-\frac{1}{x} - \frac{1}{2x^2} + C\]

Step 4: Verify by Differentiation

Differentiate the result to check if it equals the original integrand. Differentiating:\[\frac{d}{dx}\left(-\frac{1}{x} - \frac{1}{2x^2} + C\right) = \frac{1}{x^2} + \frac{1}{x^3}\]This result matches the distributed form from Step 1, confirming the integration is correct.

Key Concepts

Power Rule of Integration

The power rule of integration is an essential concept for finding indefinite integrals. It is particularly useful for integrating expressions of the form \(x^n\), where "n" is any real number except -1. The rule states that: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]Here, \(C\) represents the constant of integration, emphasized due to the indefinite nature of integration. By following this rule, you can systematically integrate each term in a polynomial expression.

• If the exponent of \(x\) is -3, for example, you apply the power rule. Increase the exponent by 1 to get \(-2\) and divide by the new exponent \(-2\), resulting in \(-\frac{1}{2x^2}\).
• The constant of integration, \(C\), ensures the solution covers all possible antiderivatives that differ by a constant.

Integrating more complex functions often requires manipulation, such as expanding or distributing to apply the power rule effectively.

Antiderivative

An antiderivative, or an indefinite integral, is a function whose derivative gives the original function. The process opposite to differentiation, finding an antiderivative, involves determining a function whose rate of change matches the given function.In this particular exercise, we focus on computing the antiderivative of \( x^{-3} (x+1) \). The initial step involves simplifying the expression by distributing \(x^{-3}\):

• Express the integrand as \(x^{-2} + x^{-3}\).
• Integrate each part separately, applying the power rule to find the antiderivative.

For \(x^{-2}\), the antiderivative is \(-\frac{1}{x}\), and for \(x^{-3}\), it is \(-\frac{1}{2x^2}\). Hence, the general antiderivative becomes \(-\frac{1}{x} - \frac{1}{2x^2} + C\). This result encapsulates the family of functions that meet the criteria of having a derivative equal to the original integrand.

Differentiation Verification

Once you find an antiderivative, it's crucial to verify its correctness by differentiation. This involves finding the derivative of your antiderivative to see if it matches the original function.For our example, the antiderivative obtained is \(-\frac{1}{x} - \frac{1}{2x^2} + C\). To verify:

• Differentiate \(-\frac{1}{x}\) to get \(\frac{1}{x^2}\).
• Differentiate \(-\frac{1}{2x^2}\) to get \(\frac{1}{x^3}\).
• The derivative of the constant \(C\) is \(0\).

The summation of these derivatives gives you \(\frac{1}{x^2} + \frac{1}{x^3}\), which matches exactly with the simplified original expression. This confirmation highlights that the integration was performed correctly and the antiderivative obtained is valid.