6、∫[( 1/x)√( x- 1)]dx = 2 √( x- 1)-∫dx/[x √( x- 1)]

=2√(x- 1)-2 arc tangent √(x- 1)+C

References:

Baidu Library-Common Integral Table, formula 17, 15.

V 1, arc length l (x) of curve x = cos3t (0 ≤ t ≤ 2π),

l(x)=4∫0→π/2 [2cos3tdt]

=-[(8/3)? Because? sint] 0→π/2

=8/3

The arc length l(y) of the curve y = sin 3t(0≤t≤2π), l(y)= l(x),

l(y)=8/3

2, if the series ∑(an)? , n: 1 →∞, convergence,

Then 0 < a < 1 or-1 < a < 0,

∴ series ∑ (- 1) n (an)/n, n: 1 →∞, absolute convergence.