Leibnitz Rule

JEE Main Mathematics · Definite Integration · 4 questions, latest first

• Let f be a real valued continuous function defined on the positive real axis such that g(x)=∫_0^x t f(t) dt. If g(x^3)=x^6+x^7,…202528 January, Shift 2 · Q18
• Let f:(0, ∞) → R and F(x)=∫_0^x t f(t) dt. If F(x^2)=x^4+x^5, then Σ_r=1^12 f(r^2) is equal to ____.20241 February, Shift 2 · Q24
• Let f be a continuous function satisfying ∫_0^t^2(f(x)+x^2) d x=4/3 t^3, ∀ t>0. Then f((π^2)/4) is equal to202310 April, Shift 2 · Q11
• If φ(x)=1/(√x) ∫_((π)/4)^x(4 √2 sin t-3 φ^′(t)) dt, x>0, then φ^′((π)/4) is equal to:202331 January, Shift 2 · Q69