A Nice Integral (2)

Calculus Level 3

Let $f(x)$ be continuous and $\displaystyle \int_{0}^{9} f(x)\, dx= 10$ . Find $\displaystyle \int_{0}^{3}xf(x^{2})\, dx$ .

10 8 7 5

1 solution

Hana Wehbi
Jun 16, 2016

Let $u = x^{2}$ , then $du= 2xdx \implies \frac{du}{2}=xdx$ $\implies \int_{0}^{3}xf(x^{2})dx= \frac{1}{2}\int_{0}^{9}f(u)du= \frac{1}{2} \times10=5$