Question

If f is continuous and 6". 16 f(x) dx = 12, find Lox xf(x2) dx.

Answer 1

$\hspace{-0.75 cm}$Here we have given $\int^{16}_{0}f(x)dx=12.$ Now let we assume \\\\\\ $I=\int^{4}_{0}x\cdot f(x^2)dx, $ The we substitute $x^2=t$ and $\frac{d}{dx}(x^2)dx=dt$\\\\\\ $2xdx=dt\Longrightarrow xdx=\frac{1}{2}dt.$ And also changing limits...\\\\\\ When $x=0.$ Then $t=x^2=0^2=0$\\\\\\ When $x=4.$ Then $t=x^2=4^2=16.$\\\\\\ So we have $I=\frac{1}{2}\int^{16}_{0}f(t)dt=\frac{1}{2}\int^{16}_{0}f(x)dx=\frac{1}{2}\times 12=\color{blue}6.$\\\\\\ $\bullet\; $ Above using definite integral formula $\int^{b}_{a}f(x)dx=\int^{b}_{a}f(t)dt.$$