Integrate $\frac{1 + x}{x^{3}} \mathrm d x$

Analysis

To solve this problem, we need to integrate the function $\int \frac{1 + x}{x^3} \, dx$. The key strategy here is to simplify the integrand by splitting the fraction before applying the power rule for integration.

Concept: The Power Rule for Integration

The power rule states that:

Step-by-Step Solution

Step 1: Split the fraction Divide each term in the numerator by the denominator ($x^3$):

Step 2: Simplify using indices Rewrite the terms using negative exponents:

Step 3: Integrate term by term Apply the power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1}$ to each term:

• For $x^{-3}$: $$\int x^{-3} \, dx = \frac{x^{-3+1}}{-3+1} = \frac{x^{-2}}{-2} = -\frac{1}{2x^2}$$

• For $x^{-2}$: $$\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} = \frac{x^{-1}}{-1} = -\frac{1}{x}$$

Step 4: Combine the results Adding the constant of integration ($k$):

Conclusion

Comparing our result with the given options, we find that it matches option B.

Exam Tip

When you see a fraction with a single term in the denominator (like $x^3$), always try to split it into separate fractions first. It makes the integration much simpler and prevents errors associated with the quotient rule (which doesn't apply to integration).