#1 Evaluate the Integral | University of Cambridge - Edubirdie

University:
University of Cambridge
Course:
0580 IGCSE | Mathematics
Academic year:

2023
Views:

462

Pages:

3
Author:

Shahbaz A.

Integration shahbaz ahmed August 2024 Evaluate the integral Z √ ax dx a2 − x4 √ ax dx a2 − x4 Solution Z (1) Let x2 = a sin θ a2 − x4 = a2 − a2 sin2 θ a2 − x4 = a2 [1 − sin2 θ] a2 − x4 = a2 [cos2 θ] √ √ a2 − x4 = a2 cos2 θ p a2 − x4 = a cos θ 1 (2) Similarly x2 = a sin θ Differentiating with respect to θ d 2 dθ (x ) = d 2 dx dx (x ) dθ d dθ (a sin θ) d = a dθ (sin θ) 2x dx dθ = a cos θ xdx = a cos θ dθ 2 (3) Putting equation (2) and equation(3) in equation (1) Z ax a √ dx = 2 a2 − x 4 Z Z Z a cos θ dθ a cos θ a ax √ dx = 2 a2 − x 4 √ Z dθ a ax dx = θ + C 2 a2 − x 4 Since x2 = a sin θ 2 Or x2 a = sin θ 2 sin−1 xa = θ Putting in the above integral Z a −1 x2 ax √ +C dx = sin 2 a a2 − x 4 3

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