Question

The value of \(\displaystyle \int \frac{1}{(x-5)^2}\,dx\) is:

• A. \(\dfrac{1}{(x-5)} + C\)
• B. \(-\dfrac{1}{(x-5)} + C\)
• C. \(\dfrac{2}{(x-5)^3} + C\)
• D. \(-2(x-5)^{-3} + C\)

Hint:

For integrals of the form (int (x-a)^n dx), always use the power rule: [ int (x-a)^n dx = frac(x-a)^n+1n+1 + C,quad n neq -1 ]

Answer 1

The Correct Option is B

Step 1: Rewrite the integrand using powers. \[ \int \frac{1}{(x-5)^2}\,dx = \int (x-5)^{-2}\,dx \] Step 2: Apply substitution. Let \[ u = x-5 \quad \Rightarrow \quad du = dx \] So the integral becomes: \[ \int u^{-2}\,du \] Step 3: Integrate using the power rule. \[ \int u^{n}\,du = \frac{u^{n+1}}{n+1} + C \quad (n \neq -1) \] Here, \(n=-2\): \[ \int u^{-2}\,du = \frac{u^{-1}}{-1} + C = -u^{-1} + C \] Step 4: Substitute back. \[ = -\frac{1}{x-5} + C \] Step 5: Final conclusion. \[ \boxed{-\dfrac{1}{(x-5)} + C} \]