It is known that f(x) is an odd function in the interval [−2p,2p] and has a period p, Prove that ∫axf(t)dt is also periodic function with the same period.
Prove that: y=∫81sin2xsin−1tdt+∫81cos2xcos−1t,where 0≤x≤2π, is the equation of a straight line parallel to the x-axis. Find the equation.
A function f is continuous for all x (and not everywhere zero) such that f2(x)=∫0xf(t)2+sintcostdt˙ Then f(x) is 211n(2x+cosx);x=0 211n(x+cosx3);x=0 211n(22+sinx);x=nπ,n∈I 2+sinxcosx+sinx;x=nπ+43π,n∈I
Let f(x) is continuous and positive for x∈[a,b],g(x) is continuous for x∈[a,b]and∫ab∣g(x)∣dx>∣∣∫abg(x)dx∣∣ STATEMENT 1 : The value of ∫abf(x)g(x)dx can be zero. STATEMENT 2 : Equation g(x)=0 has at least one root for x∈(a,b)˙