Cho \(\int\limits_0^{2017} {f\left( x \right){\rm{...

A

- Hướng dẫn giải

Đặt $t = \ln \left( {{x^2} + 1} \right),$ suy ra ${\rm{d}}t = \frac{{2x{\rm{d}}x}}{{{x^2} + 1}} \to \frac{{x{\rm{d}}x}}{{{x^2} + 1}} = \frac{{{\rm{d}}t}}{2}.$

Đổi cận: $\left\{ \begin{array}{l} x = 0 \to t = 0\\ x = \sqrt {{e^{2017}} - 1}  \to t = 2017 \end{array} \right..$

Khi đó $I = \frac{1}{2}\int\limits_0^{2017} {f\left( t \right){\rm{d}}t}  = \frac{1}{2}\int\limits_0^{2017} {f\left( x \right){\rm{d}}x}  = \frac{1}{2}.2 = 1.$