Đáp án:
\(\displaystyle\int\limits_6^9f(x)dx = 2020\)
Giải thích các bước giải:
\(\begin{array}{l}
\text{Ta có:}\\
\quad \displaystyle\int\limits_0^{\ln4}f(e^x + 5)dx = 2\\
\Leftrightarrow \displaystyle\int\limits_0^{\ln4}\dfrac{f(e^x+5).e^xdx}{e^x} = 2\\
Đặt\ u = e^x + 5\longrightarrow e^x = u - 5\\
\to du = e^xdx\\
\text{Đổi cận:}\\
x\quad\Big|\quad 0\qquad \ln4\\
\overline{u\quad\Big|\quad 6\qquad\ \ 9\quad}\\
\text{Ta được:}\\
\quad \displaystyle\int\limits_6^9\dfrac{f(u)}{u-5}du = 2\\
hay\ \ \displaystyle\int\limits_6^9\dfrac{f(x)}{x-5}dx = 2\\
\text{Mặt khác:}\\
\quad \displaystyle\int\limits_6^9\dfrac{2x+3}{x-5}f(x)dx = 4066\\
\Leftrightarrow\displaystyle\int\limits_6^9\left(2 + \dfrac{13}{x-5}\right)f(x)dx = 4066\\
\Leftrightarrow 2\displaystyle\int\limits_6^9f(x)dx + 13\displaystyle\int\limits_6^9\dfrac{f(x)}{x-5}dx = 4066\\
\Leftrightarrow 2\displaystyle\int\limits_6^9f(x)dx + 13\cdot 2 = 4066\\
\Leftrightarrow \displaystyle\int\limits_6^9f(x)dx = 2020
\end{array}\)
