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Giải bài tập 4.35 trang 37 SGK Toán 12 tập 2 - Cùng khám phá

Cho \(F(x)\) là một nguyên hàm của hàm số \(f(x) = \frac{2}{x}\), biết \(F(1) = 2\). Giá trị của \(F(3)\) bằng: A. \(2 + 2\ln 3\) B. \(2 + \ln 3\) C. \(2 - 2\ln 3\) D. \(2 - \ln 3\)
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Đề bài

Cho \(F(x)\) là một nguyên hàm của hàm số \(f(x) = \frac{2}{x}\), biết \(F(1) = 2\). Giá trị của \(F(3)\) bằng: A. \(2 + 2\ln 3\) B. \(2 + \ln 3\) C. \(2 - 2\ln 3\) D. \(2 - \ln 3\)

Phương pháp giải - Xem chi tiết

- Tìm nguyên hàm của hàm số \(f(x) = \frac{2}{x}\). - Áp dụng điều kiện \(F(1) = 2\) để tìm hằng số tích phân. - Tính giá trị của \(F(3)\).

Lời giải chi tiết

\(F(x) = \int f (x){\mkern 1mu} dx = \int {\frac{2}{x}} {\mkern 1mu} dx = 2\ln |x| + C\) Vì \(x > 0\), ta có: \(F(x) = 2\ln x + C\) Áp dụng điều kiện \(F(1) = 2\) \(F(1) = 2\ln 1 + C = C = 2\) Do đó, \(F(x) = 2\ln x + 2\). \(F(3) = 2\ln 3 + 2\) Chọn A.

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