Question

if ( f ) is continuous and ( int_{0}^{49} f(x) d x=8 ), find ( int_{0}^{7} x fleft(x^{2}
ight) d x ).

Explanation:

Step1: Substitution

Let \(u = x^{2}\), then \(du=2xdx\), and \(xdx=\frac{1}{2}du\).
When \(x = 0\), \(u = 0\); when \(x = 7\), \(u=49\).

Step2: Rewrite the integral

\(\int_{0}^{7}xf(x^{2})dx=\frac{1}{2}\int_{0}^{49}f(u)du\)

Step3: Use the given value

Since \(\int_{0}^{49}f(x)dx = 8\) (and \(\int_{0}^{49}f(u)du=\int_{0}^{49}f(x)dx\) by the property of definite - integrals that the variable of integration is a dummy variable), then \(\frac{1}{2}\int_{0}^{49}f(u)du=\frac{1}{2}\times8\)

Answer:

\(4\)