Question
If _ 0 ^ 1 4 ^ -1 (1-2 x+4 x^ 2 ) d x= atan ^ -1 (2)- blog _ e (5), where a , b N , then (2 a + b ) is equal to \_\_\_\_.
JEE Main
Mathematics
Definite Integration
2026
JEE Main 2026 (21 January Shift 2)
JEE Main Mathematics Question (2026) — Solution
Options
- A. A
- B. B
- C. C
- D. D
Answer
A. A
Step-by-step solution
Let I= _0^1 ^ -1 (1-2 x+4 x^2 ) d x I= _0^1 ( ^ -1 (2 x-1)- ^ -1 (2 x) ) d x Applying king I= _0^1 (- ^ -1 (2 x-1)+ ^ -1 (2 x-2) ) d x From (1) and (2) 2 I= _0^1 ( ^ -1 (2 x-2)- ^ -1 (2 x) ) d x = _0^1 ^ -1 (2 x-2) d x- _0^1 ^ -1 (2 x) d x Applying King = _0^1 ^ -1 (-2 x) d x- _0^1 ^ -1 (2 x) d x = _0^1 ( - ^ -1 (2 x) ) d x- _0^1 ^ -1 (2 x) d x = _0^1 ( -2 ^ -1 (2 x) ) d x = -2 _0^1 ( ^ -1 2 x ) 1 d x By parts = -2 [ (x ^ -1 2 x )_0^1+ _0^1 2 x 1+4 x^2 d x ] Let 1+4 x ^2= t 8 xdx = dt = -2 [ ^ -1 2+ 1 4 _1^5 dt t ] = -2 ^ -1 2- 1 2 n 5 2 I =2 ^ -1 2- 1 2 n 5 4 I =4 ^ -1 2- n 5 2 a + b =8+1=9
Related: Mathematics — Definite Integration · All PYQ Banks