AP EAMCET202316 May 2023Evening ShiftMathematicsDefinite IntegrationActual
Assertion (A): ₀^ 2 ( ^6 x+ ^6 x ) d x lies in the interval ( 8 , 2 ) Reason (R): ^6 x+ ^6 x is a periodic function with period 2
Options
- ABoth A and R are true and R is the correct explanation of A
- BBoth A and R are true but R is not the correct explanation of A
- CA is true, R is false
- DA is false, R is true
Correct answer
B. Both A and R are true but R is not the correct explanation of A
Step-by-step solution
Let I= ₀^ / 2 ( ^6 x+ ^6 x ) d x aligned & I= ₀^ / 2 ( ( ^2 x )^3+ ( ^2 x )^3 ) d x & = ₀^ 2 (1- 3 4 ( 2 x)^2 ) d x aligned Since 1 4 1- 3 4 ( 2 x)^2 1 for x [0, 2 ] ₀^ / 2 1 4 d x ₀^ / 2 (1- 3 4 ( 2 x)^2 ) d x ₀^ / 2 1 d x 8 < I < 2 Assertion is correct and f (x+ 2 )= ^6 (x+ 2 )+ ^6 (x+ 2 )= ^6 x+ ^6 x Period of ^6 x+ ^6 x is 2 since reason R is correct.