AP EAMCET2004MathematicsDefinite Integration
If f(x)= 1 x^2 ₃^x (2 t-3 f^ (t) ) d t , then f^ (3) is equal to
Options
- A-1 2
- B-1 3
- C1 2
- D1 3
Correct answer
C. 1 2
Step-by-step solution
Given that, f(x)= 1 x^2 ₃^x (2 t-3 f^ (t) ) d t On differentiating both sides w.r.t. x , we get aligned & f^ (x)= 1 x^2 d d x ( ₃^x (2 t-3 f^ (t) d t ) . & + ₃^x (2 t-3 f^ (t) ) d t d d x ( 1 x^2 ) aligned aligned & f^ (x)= 1 x^2 (2 x-3 f^ (x) ) &- 2 x^3 ₃^x (2 t-3 f^ (t) ) d t aligned At x=3 , aligned f^ (3) & = 1 3^2 (6-3 f^ (3) )-0 4 f^ (3) 3 & = 2 3 f^ (3)= 1 2 aligned