Tích phân \(\int\limits^4_2x.\ln\left(x-1\right)\text{d}x\) bằng
\(7,5\ln3+4\). \(\dfrac{15\ln3}{2}-4\). \(\dfrac{15\ln3}{2}+2\). \(7,5\ln3-2\). Hướng dẫn giải:\(\int\limits^4_2x.\ln\left(x-1\right)\text{d}x=\int\limits^4_2\ln\left(x-1\right)\left(\dfrac{x^2-1}{2}\right)'\text{d}x=\dfrac{\left(x^2-1\right)\ln\left(x-1\right)}{2}|_2^4-\int\limits^4_2\dfrac{x^2-1}{2}\left(\ln\left(x-1\right)\right)'\text{d}x\)
\(=\dfrac{15\ln3}{2}-\int\limits^4_2\dfrac{x^2-1}{2}.\dfrac{1}{x-1}\text{d}x=\dfrac{15\ln3}{2}-\int\limits^4_2\dfrac{x+1}{2}\text{d}x\)
\(=\dfrac{15\ln3}{2}-\dfrac{\left(x+1\right)^2}{4}|_2^4=\dfrac{15\ln3}{2}-4\)
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