`Ify=int_0^xf(t)sin{k(x-t)dt ,t h e np rov et h a t(dt^2y)/(dx^2)+k^2y=kf(x)dot`

If int_0^ycost^2dt=int_0^(x^2)(sint)/t dx ,t h ep rov et h a t(dy)/(dx)=(2sinx^2)/(xcosy^2)

Iff(x)=int_1^x(logt)/(1+t+t^2)dxAAxlt=1,t h e np rov et h a tf(x)f(1/x)dot

If y=sqrt(x+sqrt(y+sqrt(x+sqrt(y+oo)))),t h e np rov et h a t(dy)/(dx)=(y^2-x)/(2y^3-2x y-1)

Given I_m=int_1^e(logx)^mdx ,t h e np rov et h a t(I_m)/(1-m)+m I_(m-2)=e

IfI_n=int_0^1x^n(tan^(-1)x)dx ,t h e np rov et h a t (n+1)I_n+(n-1)I_(n-2)=-1/n+pi/2

If f(x+2a)=f(x-2a),t h e np rov et h a tf(x)i sp e r iod i cdot

If y=sqrt(sinx+sqrt(sinx+sqrt(sinx+...... tooo))),p rov et h a t(dy)/(dx)=(cosx)/(2y-1)

If x int_0^xsin(f(t))dt=(x+2)int_0^x tsin(f(t))dt ,w h e r ex >0, then show that f^(prime)(x)cotf(x)+3/(1+x)=0.

If int_(0)^(x^(2)(1+x))f(t)dt=x , then the value of f(2) is.

If f(x)=int_(2)^(x)(dt)/(1+t^(4)) , then