Cho hàm số \(f(x)=x^{2}+4 x+2\).
Biết \(\int_{0}^{3}\left[2 f(x)+3^{-x}\right] d x=a+\frac{c}{d \ln 3}\), trong đó \(\frac{c}{d}\) là phân số tối giản. Tính \(T=a-c-d\)
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\(\int_{0}^{3}\left[2 f(x)+3^{-x}\right] d x=2 \int_{0}^{3} f(x) d x+\int_{0}^{3} 3^{-x} d x=\left.2\left(x^{2}+4 x+2\right)\right|_{0} ^{3}+\int_{0}^{3}\left(\frac{1}{3}\right)^{x} d x=66+\left.\frac{\left(\frac{1}{3}\right)^{x}}{\ln \frac{1}{3}}\right|_{0} ^{3}\)
\(=66-\frac{26}{27 \ln \frac{1}{3}}=66+\frac{26}{27 \ln 3}\)
Do đó \(a=66, c=26, d=27\). Vậy \(T=a-c-d=66-26-27=13\).
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