Cho hàm số \(f\left( x \right)\) liên tục trên \(\mathbb{R}\) và thỏa \(\int\limits_{-2}^{2}{f\left( \sqrt{{{x}^{2}}+5}-x \right)\text{d}x}=1,\int\limits_{1}^{5}{\frac{f\left( x \r...

* Hướng dẫn giải

Đặt: \(t=\sqrt{{{x}^{2}}+5}-x\Rightarrow x=\frac{5-{{t}^{2}}}{2t}\Rightarrow \text{d}x=-\left( \frac{1}{2}+\frac{5}{2{{t}^{2}}} \right)\text{d}t\).

Ta có: \(1=\int\limits_{1}^{5}{f\left( t \right)}\left( \frac{1}{2}+\frac{5}{2{{t}^{2}}} \right)\text{d}t=\frac{1}{2}\int\limits_{1}^{5}{f\left( t \right)}\text{d}t+\frac{5}{2}\int\limits_{1}^{5}{\frac{f\left( t \right)}{{{t}^{2}}}\text{d}t}\)

\(\Rightarrow \frac{1}{2}\int\limits_{1}^{5}{f\left( t \right)}\text{d}t=1-\frac{5}{2}\int\limits_{1}^{5}{\frac{f\left( t \right)}{{{t}^{2}}}\text{d}t}=1-\frac{5}{2}.3=-\frac{13}{2}\)

\(\Rightarrow \int\limits_{1}^{5}{f\left( t \right)}\text{d}t=-13\)