L'intégrale de cos(5x) est (sin(5x))/5

1)
Poser x = 2.sin(t)
dx = 2.cos(t).dt
4 - x² = 4 - 4.sin²t = 4(1-sin²t) = 4.cos²t

S V(4-x²).dx = S 2.cos(t).2.cos(t).dt = 4.S cos²(t).dt
= 4.S (1/2).(1+cos(2t)).dt
= 2.S (1+cos(2t)).dt
= 2.(t + (1/2).sin(2t)) + C
= 2.(arcsin(x/2) + (1/2).sin(2arcsin(x/2))) + C
= 2.(arcsin(x/2) + (1/2).2sin(sin(arcsin(x/2)).cos(arcsin(x/2)) + C
= 2.(arcsin(x/2) + (1/2).2sin(arcsin(x/2)).cos(arcsin(x/2)) + C
= 2.(arcsin(x/2) + (1/2).x.V(1-(x/2)²) + C   (avec V pour racine carrée).
= 2.arcsin(x/2) + x.V(1-(x/2)²) + C
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2)
Poser sin(5x) = t
5.cos(5x). dx = dt

S cos(5x).dx = (1/5).S dt = (1/5).t + C = (1/5).sin(5x) + C
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Sauf distraction.