`Iflambda=int_0^(pi/2)costhetaf(sintheta+cos^2theta)dtheta` `a n dI_2=int_0^(pi//2)sin2thetaf(sintheta+cos^2theta)dtheta,t h e n` `I_1=-2I_2` (b) `I_1=I_2` `2I_1=I_2` (d) `I_1=-I_2`

int_0^(pi//2)(theta/sintheta)^2d theta=

Evaluate: int_0^(pi//2)(costheta)/((1+sintheta)(2+sintheta)) d theta

Let f beintegrable over [0,a] for any real value of a . If I_(1)=int_(0)^(pi//2)cos theta f(sin theta +cos^(2) theta) d theta and I_(2)=int_(0)^(pi//2) sin 2 theta f(sin theta+cos^(2) theta) d theta , then

The value of I(n) =int_0^pi(sin ^2n theta)/(sin^2theta) d theta is (AA n in N)

If I_1=int_0^(3pi) (cos^2x) dx and I_2 =int_0^pi (cos^2x) dx then (a) I_1=I_2 (2) I_1=2 I_2 (3) I_1=5I_2 (4) I_1=3I_2

I_1=int_0^(pi/2)ln(sinx)dx ,I_2=int_(-pi/4)^(pi/4)ln(sinx+cosx)dxdot Then (a) I_1=2I_2 (b) I_2=2I_1 I_1=4I_2 (d) I_2=4I_1

If I_n = int_0^(pi/2) (sin^2 nx)/(sin^2 x) dx , then

Column I, Column II If f(x) is an integrable function for x in [pi/6,pi/3]a n d I_1=int_(pi/6)^(pi/3)sec^2thetaf(2sin2theta)dtheta,a n d I_2=int_(pi/6)^(pi/3)"c o s e c"^2thetaf(2sin2theta)dtheta,t h e n(I_1)/(I_2)= , p. 3 If f(x+1)=f(3+x)AAx , and the value ofint_a^(a+b)f(x)dxi sin d e p e n d e n tofa , then the value of b can be, q. 1 The value of int_0^1(tan^(-1)[x^2])/(tan^(-1)[x^2]+tan^(-1)[25+x^2-10 x]) , r. 2 (d)IfIint_0^2sqrt(x+sqrt(x+sqrt(x+oo)))dx(w h e r ex >0),t h e n[I] is equal to (where [.] denotes the greatest integer function), s. 4

If I_1=int_0^pixf(sin^3x+cos^2x)dxa n d I_2=int_0^(pi/2)f(sin^3x+cos^2x)dx ,t h e nr e l a t eI_1a n dI_2

If I_1=int_0^pixf(sin^3x+cos^2x)dxand I_2=int_0^(pi/2)f(sin^3x+cos^2x)dx , then relate I_1 and I_2