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Cho (fleft( x right) = {x^2}ln x) và (gleft( x right) = xln x). Tính (f'left( x right)) và (int {gleft( x right)dx} ).

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Đề bài

Cho \(f\left( x \right) = {x^2}\ln x\) và \(g\left( x \right) = x\ln x\). Tính \(f'\left( x \right)\) và \(\int {g\left( x \right)dx} \).

Phương pháp giải - Xem chi tiết

Sử dụng các công thức tính đạo hàm và nguyên hàm để tính \(f'\left( x \right)\). Từ đó, viết biểu thức \(g\left( x \right) = x\ln x\) theo \(f'\left( x \right)\) và tính \(\int {g\left( x \right)dx} \)

Lời giải chi tiết

Ta có \(f'\left( x \right) = \left( {{x^2}\ln x} \right)' = 2x\ln x + {x^2}.\frac{1}{x} = 2x\ln x + x = 2g\left( x \right) + x\) Suy ra \(g\left( x \right) = \frac{1}{2}\left[ {f'\left( x \right) - x} \right] \Rightarrow \int {g\left( x \right)dx}  = \frac{1}{2}\int {\left[ {f'\left( x \right) - x} \right]dx}  = \frac{1}{2}\left[ {f\left( x \right) - \frac{{{x^2}}}{2}} \right] + C\),  tức là \(\int {x\ln xdx}  = \frac{1}{2}\left( {{x^2}\ln x - \frac{{{x^2}}}{2}} \right) + C\)

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