Question

if f is continuous and the integral from [0,9] f(x)dx=4 find the integral [0,3] xf(x^2)dx

Answer 1

\[\int\limits_0^9 f(x)dx=4\]\[\int\limits_0^3 xf(x^2)dx=\frac{1}{2}\int\limits_0^3 f(x^2)d(x^2)\quad,\quad y=x^2\]\[\int\limits_0^3 xf(x^2)dx=\frac{1}{2}\int\limits_0^3 f(x^2)d(x^2)=\frac{1}{2}\int\limits_0^9 f(y)dy=\frac{1}{2}\cdot 4=2\]