Prob 3
sin8 x ?? cos8 x = (sin4 x ?? cos4 x)(sin4 x + cos4 x) = (sin2 x ?? cos2 x)(sin4 x + cos4 x)
1 ?? 2 sin2 x cos2 x = (sin2 x + cos2 x)2 ?? 2 sin2 x cos2 x = (sin4 x + cos4 x)
Thus sin8 x??cos8 x
1??2 sin2 x cos2 x = sin2 x ?? cos2 x = ??cos 2x. Thus
R sin8 x??cos8 x
1??2 sin2 x cos2 xdx = ??sin 2x
2 + C
Prob 4
for the function f(x) = jxj
f0(x) =
(
??1 ??1 x0 hence x = =2 is point of minima and minimum value is f(=2) = 3.
f00(sin??1(1=3)) = ??2??4
p
2
30
f00(??1) = ??7 < 0
f(1) = 1=3 ; f(??1) = 3 are minimum and maximum values of the function respectively.
Prob 9
x2+1
x+1 ?? ax ?? b = (1??a)x2??x(a+b)+(1??b)
x+1
given limx!1
(1??a)x2??x(a+b)+(1??b)
x+1 = 0
thus degree(num)
raju , 11 Years ago
Grade 12
