The function \(f\) is continuous and \(\int\limits_0^8 f(u)  du = 10\). What is the value of \(\int\limits_1^3 xf(x^2-1)  dx \) ?

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$$ \int\limits_1^3 xf(x^2-1)  dx $$

Use the following substitutions:

$$ u=x^2-1 \hskip{2em} du=2x  dx \hskip{2em} dx=\frac{1}{2x}du$$ $$ \int\limits_0^8 xf(u)\cdot \frac{1}{2x}  du =\frac{1}{2}\int\limits_0^8 f(u)  du $$ $$ = \frac{1}{2}(10) =\boxed{5}$$