Theory question 111
Integration(a) Using the substitution U = 5 − x2, evaluate ∫12 x/√(5 − x2) dx
(b) If y = px2 + qx, dy/dx = 7 and d2y/dx2 = 6, find the values of p and q.
Show the model answer
Model answer
(a) U = 5 − x², dU = −2x dx; x = 1 → U = 4, x = 2 → U = 1
Integral = ∫41 −½U−½ dU = ½∫14 U−½ dU = [√U] from 1 to 4 = 2 − 1 = 1
(b) dy/dx = 2px + q, d²y/dx² = 2p = 6, so p = 3
Then 6x + q = 7, so q = 7 − 6x; if the condition is at x = 1 (as intended), q = 1
Explanation
Change the limits when you substitute; d²y/dx² gives p straight away.