Cho hàm số: . Tích phân \(\int\limits_{\frac{1}{e}}^{e}{\frac{f(3\ln x+4)}{x}}dx\) bằng

* Hướng dẫn giải

Tích phân \(\int\limits_{\frac{1}{e}}^{e}{\frac{f(3\ln x+4)}{x}}dx\). Đặt \(3\ln x+4=t\Rightarrow \frac{dx}{x}=\frac{1}{3}dt\)

\(\begin{align} & \int\limits_{\frac{1}{e}}^{e}{\frac{f(3\ln x+4)}{x}}dx=\int\limits_{1}^{7}{f(t)}.\frac{1}{3}dt=\frac{1}{3}\int\limits_{1}^{5}{f(t)}dt+\frac{1}{3}\int\limits_{5}^{7}{f(t)}dt \\ & =\frac{1}{3}\int\limits_{1}^{5}{(3t+2)}dt+\frac{1}{3}\int\limits_{5}^{7}{(4-6{{t}^{2}})}dt=-128 \\ \end{align}\)