If `varphi(x)=1/(1+e^(-x))a n dS=varphi(5)+varphi(4)+varphi(3)++varphi(-3)+varphi(-4)+varphi(-5)` , then the value of `S` is. a. 5 b. 11/2 c. 6 d. 13/2

Evaluate: int_(pi/4)^((3pi)/4) varphi/(1-sinvarphi)d varphi

If y+x(dy)/(dx)=x(varphi(xy))/(varphi'(xy)) then varphi(xy) is equal to

Evaluate: int(2sin2varphi-cosvarphi)/(6-cos^2varphi-4\ sinvarphi)\ dvarphi

If varphi(x) is differentiable function AA x in R and a in R^(+) such that varphi(0)=varphi(2a),varphi(a)=varphi(3a)and varphi(0)!=varphi(a) then show that there is at least one root of equation varphi'(x+a)=varphi'(x)in(0,2a)

Let int e^(x){f(x)-f'(x)}dx=varphi(x) .Then int e^(x)f(x)dx is varphi(x)=

(tan(theta+varphi)+tan(theta-varphi))/(1-tan(theta+varphi)tan(theta-varphi))=tan2 theta

Prove that the product of matrices [cos^(2)theta cos theta sin theta cos theta sin theta s in^(2)theta] and [cos^(2)varphi cos varphi sin varphi cos varphi sin varphi s in^(2)varphi] is the null matrix,when theta and varphi differ by an odd multiple of (pi)/(2) .