Question

int (sqrt (x^2-9))/ x x>3

Answer 1

it's copied from a question before

let x=3 sin(a)
dx=3cos(a)da
a = arcsin(x/3)

int (9 sin^2(a) 3 cos(a) da / sqrt(9-9sin^2(a)))
=int (27 sin^2(a)cos(a)da/sqrt(9cos^2(a)) **note:[1-sin^2(a)=cos^2(a)]
= int (27 sin^2(a)cos(a)da/3cos(a))
= int (9 sin^2(a)da) **note:[sin^2(a)=sin(a)*sin(a)= 0.5(cos(0)-cos(2a))
= int (4.5(1-cos(2a))da)
= int (4.5da) - int(4.5cos(2a) da)
= 4.5a - 2.25sin(2a) +c
=4.5arcsin(x/3) - 4.5sin(a)cos(a) + c *note:[refer to the lets;; and sin(2a)=2sin(a)cos(a)]
=4.5arcsin(x/3) - 4.5(x/3)sqrt(9-x^2)/3 +c
=4.5arcsin(x/3) - 0.5xsqrt(9-x^2)/9 +c