Mathematics 2019 JAMB Past Questions - Question 21: Integrate 1-x/x3 with respect to x

Correct Answer

B

Explanation

To integrate \( \frac{1-x}{x^3} \) with respect to \( x \), we can split the fraction into two separate fractions:\[ \frac{1}{x^3} - \frac{x}{x^3} \]Now, we can integrate each term separately:\[ \int \frac{1}{x^3} \, dx - \int \frac{x}{x^3} \, dx \]The first integral can be solved using the power rule for integration:\[ \int \frac{1}{x^3} \, dx = \int x^{-3} \, dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C_1 \]For the second integral, we can simplify the fraction first:\[ \int \frac{x}{x^3} \, dx = \int x^{-2} \, dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C_2 \]So, the final result of the integration is:\[ -\frac{1}{2x^2} - \frac{1}{x} + C \]where \( C \) is the constant of integration.