Let g:$[0,\infty) \to\mathbb{R}$ be continuous function satisfying $\int_{0}^{x^2(1+x)} g(t)dt=x , for all x\in[0,\infty)$. Then g(2) is equal to ?
given : $\int_{0}^{x^2(1+x)} g(t)dt=x $ differentiating both sides with respect to x
$\frac{d}{dx}$$\int_{0}^{x^2(1+x)} g(t)dt=1 $ . Let $\int g(x)dt=G(x)$
$\Rightarrow$ $\frac{d}{dx}$[G($x^2(1+x))-G(0)$]=1
$\Rightarrow$$[g(x^2(1+x)$) (2x(1+x)+x^2)] - 0 =1
put x=1 $\Rightarrow$ g(2)=1/5
please correct me if i am wrong
