first write

sin 3x = 3 sin x - 4 sin³ x

so integral becomes

= 1/ 4 sinx - 4sin ³ x

take 4 sin x common from denom

= 1 / 4sin x ( 1 - sin² x )

= 1 / 4 sinx X cos ² x

= sec²x / 4 sin x

= 1 + tan²x / 4 sin x

= 1/4 [ 1 / sinx + tan²x / sin x ]

= 1/4 [ cosec x + sinx / cos² x ]

= 1/4 [ cosec x + sec x X tanx ]

so the ans is

1/4 [ log ( cosec x - cot x ) + sec x + C ]

ques 44



as the door is in equilibrium

net torque acting on it is zero



so let the force by the hinge be N

now take torque bout any of the hinge



we will get



Mg x perpendicular distance from hinge - N x perpendicular distance from hinge = 0

thus u can find N which is the force exerted by the hinge



ques 77



let the radius of hemispherical bowl be R and that of spherer be r

so initially it has some potential energy

using energy conservationn,

mgR = 1/2 x m x v² + 1/2 x I x ²

mgR = 1/2 x m x v² + 1/2 x 2/5 x m x r² x v² / r²

simplifying the above we get

10gR = 7v² -----------   1

now let the normal force be N

using newtons laws of motion,

N - mg = mv² / R

7N -7mg = 7mv² / R

using the 1st equation ,

7N- 7mg = 10mg

or N = 17mg / 7