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Giải bài tập 4.5 trang 10 SGK Toán 12 tập 2 - Cùng khám phá

Biết \(F(x) = {e^x} + {x^2}\) là một nguyên hàm của hàm số \(f(x)\) trên \(\mathbb{R}\) và hàm số \(f'(x)\) liên tục trên \(\mathbb{R}\). Tìm \(\int {f'} (x){\mkern 1mu} dx\).
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Đề bài

Biết \(F(x) = {e^x} + {x^2}\) là một nguyên hàm của hàm số \(f(x)\) trên \(\mathbb{R}\) và hàm số \(f'(x)\) liên tục trên \(\mathbb{R}\). Tìm \(\int {f'} (x){\mkern 1mu} dx\).

Phương pháp giải - Xem chi tiết

Tính đạo hàm của \(F(x)\) để tìm hàm số \(f(x)\), sau đó tích phân \(f'(x)\) để tìm kết quả.     

Lời giải chi tiết

Đạo hàm của \(F(x)\): \(f(x) = F'(x) = {e^x} + 2x\) Do đó: \(\int {f'} (x){\mkern 1mu} dx = f(x) + C = {e^x} + 2x + C\)

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