If ₀^ k 1 1 + 4^x dx = ₂ ( 4 3 ) , then k = _____

Step-by-step solution

Let I = ₀^ k 1 1 + 4^x dx Substitute 4^x = t x 4 = t dx = dt t 4 When x = 0 , t = 1 . When x = k , t = 4^k . I = ₁^ 4^k 1 1 + t dt t 4 = 1 4 ₁^ 4^k ( 1 t - 1 1 + t ) dt I = 1 4 [ t - (1 + t) ]₁^ 4^k = 1 4 [ ( t 1 + t ) ]₁^ 4^k I = 1 2 2 ( ( 4^k 1 + 4^k ) - ( 1 2 ) ) = 1 2 2 ( 2 4^k 1 + 4^k ) Given I = ₂ ( 4 3 ) = (4/3) 2 Equating the two expressions: 1 2 2 ( 2 4^k 1 + 4^k ) = (4/3) 2 ( 2 4^k 1 + 4^k ) = 2 ( 4 3 ) = ( 16 9 ) 2 4^k 1 + 4^k = 16 9 18 4^k = 16 + 16 4^k 2 4^k = 16 4^k = 8 2^ 2k = 2^3 2k = 3 k = 3 2 Answ