`T h e m a x i m u m v a l u e o f4sin^2x+3cos^2x+sinx/2+cosx/2i s(A)4+sqrt(2)(C )9`

I fx+1/x=2,t h e p r i n c i p a l v a l u e o f sin^(-1)x is

The minimum value of the function f(x)=(sinx)/(sqrt(1-cos^2x))+(cosx)/(sqrt(1-sin^2x))+(tanx)/(sqrt(sec^2x-1))+(cotx)/(sqrt(c o s e c^2x-1)) whenever it is defined is

The minimum value of the function f(x)=(sinx)/(sqrt(1-cos^2x))+(cosx)/(sqrt(1-sin^2x))+(tanx)/(sqrt(sec^2x-1))+(cotx)/(sqrt(c o s e c^2x-1)) whenever it is defined is

Ifint_(-pi/4)^((3pi)/4)((e^(pi/4)dx)/((e^x+e^(pi/4))(sinx+cosx))=kint_(-pi/2)^(pi/2)secx dx ,t h e nt h ev a l u eofki s 1/2 (b) 1/(sqrt(2)) (c) 1/(2sqrt(2)) (d) -1/(sqrt(2))

E v a l u a t e:int(x^4+x^2+1)/(x^2-x+1)dxdot

If I_n=intsin^n xdxn in N , t h e n5I_4-6I_6 is equal to s in x(cosx)^5+c b. sin2x cos2x+c c. (s in2x)/8(cos^2 2x+1-2cos2x)+c d. (s in2x)/8(cos^2 2x+1+2cos2x)+c

Recall that sinx + cosx =u (say) and sin x cosx =v (say) are connected by (sinx +cosx)^(2) = sin^(2)x + cos^(2)x+2sin cosx rArr u^(2) = 1+2v rArr v=(u^(2)-1)/(2) It follows that any rational integral function of sinx + cosx , and sinx cosx i.e., R(sinx + cosx, sinx cosx) , or in our notation R(u,v) can be transformed to R(u, (u^(2)-1)/2) . Thus, to solve an equation of the form R(u,v)=0 , we form a polynomial equation in u and than look for solutions. The solution set of sinx + cosx -2sqrt(2) sin x cosx=0 is completely described by

Recall that sinx + cosx =u (say) and sin x cosx =v (say) are connected by (sinx +cosx)^(2) = sin^(2)x + cos^(2)x+2sin cosx rArr u^(2) = 1+2v rArr v=(u^(2)-1)/(2) It follows that any rational integral function of sinx + cosx , and sinx cosx i.e., R(sinx + cosx, sinx cosx) , or in our notation R(u,v) can be transformed to R(u, (u^(2)-1)/2) . Thus, to solve an equation of the form R(u,v)=0 , we form a polynomial equation in u and than look for solutions. The solution set of sinx + cosx -2sqrt(2) sin x cosx=0 is completely described by

if int e^(2x)sin x cos xdx=(u)/(8)e^(2x)+c then u