I have two results:
$\int_2^3f(x)dx=5$
and
$\int_1^2 xf(x^2-1)dx=8$
I need to calculate: $$\int_0^2 f(x)dx$$
I have no idea about using the previous results, any hint?
2026-03-31 18:57:47.1774983467
How evaluating $\int_0^2 f(x)dx$ from $\int_2^3 f(x)dx$ and $\int_1^2xf(x^2-1)dx$?
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Hint. One may perform a change of variable, $u=x^2-1$, $du=2xdx$, giving $$ \int_1^2 xf(x^2-1)dx=\frac12\int_1^2 2xf(x^2-1)dx=\frac12\int_0^3 f(u)du=8 $$ then one may write $$ \int_0^2 f(x)dx=\int_0^3 f(x)dx+\int_3^2 f(x)dx=\int_0^3 f(x)dx-\int_2^3 f(x)dx $$ Can you take it from here?