Question

If \(\int x(a-x)^{9}dx=\frac{[f(x)]^{11}}{11}-a\frac{[f(x)]^{10}}{10}+c\), where \(c\) is a constant of integration, then \(f'(x)=\cdots\)

• A. \(a\)
• B. \(-1\)
• C. \(-a\)
• D. \(1\)

Hint:

Substitute t equal to a minus x, integrate, and compare with the given form to identify f(x).

Answer 1

The Correct Option is B

Step 1: Idea
The given answer has the form $\dfrac{t^{11}}{11}-a\dfrac{t^{10}}{10}$ with $t=f(x)$.

Step 2: Differentiate the Form
$\dfrac{d}{dx}$ gives $f'(x)\,[f^{10}-af^{9}]=f'(x)f^{9}(f-a)$.

Step 3: Match
This must equal $x(a-x)^{9}$. Taking $f=a-x$ gives $f'(a-x)^{9}(-x)$, which requires $f'=-1$.

Final Answer:
Matching with the integrand gives f'(x) equal to minus 1. \[ \boxed{\text{(B) }-1} \]