Let y=sin^-1 x

siny =x

differentiate both sides with respect to x

cos y dy/dx =1

sqrt(1-sin^2 y) dy/dx=1

sqrt(1-x^2) dy/dx=1

dy/dx = 1/sqrt(1-x^2)

2)

y=sin^-1 x

sin y = x

cos y dy/dx = 1

dy/dx = 1/cos y

Let sin^-1(x) = u

x = sin u......(i)

Differentiate

dx/du = cos(u) = sqrt(1- sin^2(u)) = sqrt(1-x^2) ....(ii)

and y =u

Differentiate

dy/dx = du/dx = 1/cosy .....(ii)

From (i) and(ii)

dy/dx = 1/1/√(1-x^2)

Hence ((d\dx) sin^-1 x) = 1/√(1-x^2) .............Ans

int sin^4x = a million/4 int (2 sin^2x)^2 dx = a million/4 int [ a million- cos2x]^2 dx =a million/4 int [ a million - 2 cos2x + cos^2 (2x) ] dx = a million/4 x - a million/4 sin2x + a million/4 int [ a million+ cos4x]/2 dx = a million/4x- a million/4sin2x + a million/8 x + a million/32 sin4x + C = 3/8x - a million/4 sin2x + a million/32 sin4x + C