for every continuous function prove that $\int_0^{x} (t^3)f(t^2)dt = 0.5 \int_0^{x^2} tf(t)dt$

Question

I thought about the Fundamental theorem of calculus but dont really know how to advance from there.

Hint: the derivative of LHS is equal to the derivative of RHS, and LHS is equal to RHS for $x =0$. — Botnakov N., Oct 23 '21 at 13:10
In the integral in the left side, use the substitution $t^2=u$ and then interchange $u$ with $t$ after the substitution. — Laxmi Narayan Bhandari, Oct 23 '21 at 13:14
I cannot help you till you show more effort. My answer was downvoted. Why not show exactly what you've done with the FTC? — Deepak, Oct 23 '21 at 13:23

score 2 · Answer 1 · answered Oct 23 '21 at 14:46

2

I think it's just a variable change $u = t^2$, then $\dfrac{1}{2}du = t dt$, when $t=0$, $u=0$ and and when $t=x$, $u = x^2$.

$\int_0^{x} (t^3)f(t^2)dt = \int_0^{x} (t^2)f(t^2)tdt = \dfrac{1}{2}\int_0^{x^2} uf(u)du.$

answered Oct 23 '21 at 14:46

Ilovemath

1 Answers1