Theory question 109
Integration(a) Given that ∫−1m (2x2 − x − 3) dx = −9/2, where m is an integer, find the value of m.
(b) Solve 2(log3 x − 1) = log3 x and y = √x + 1 simultaneously
Show the model answer
Model answer
(a) [2x³/3 − x²/2 − 3x] from −1 to m = (2m³/3 − m²/2 − 3m) − 11/6 = −9/2.
Multiply by 6: 4m³ − 3m² − 18m + 16 = 0. m = 2 is a root (32 − 12 − 36 + 16 = 0); the other roots of 4m² + 5m − 8 = 0 are not integers. m = 2.
(b) 2log₃ x − 2 = log₃ x gives log₃ x = 2, so x = 9.
y = √9 + 1 = 4. Solution: x = 9, y = 4.
Explanation
In (b) collect the log terms first; x must be positive.