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Giải bài 5 trang 14 sách bài tập toán 12 - Chân trời sáng tạo

Cho hàm số (fleft( x right)) có đạo hàm (f'left( x right) = frac{{sqrt x - 1}}{x},x > 0). Tính giá trị của (fleft( 4 right) - fleft( 1 right)).

💖Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Cho hàm số \(f\left( x \right)\) có đạo hàm \(f'\left( x \right) = \frac{{\sqrt x  - 1}}{x},x > 0\). Tính giá trị của \(f\left( 4 \right) - f\left( 1 \right)\).

Phương pháp giải - Xem chi tiết

Sử dụng các công thức: • \(\int\limits_a^b {f\left( x \right)dx}  = F\left( b \right) - F\left( a \right)\). • \(\int {{x^\alpha }dx}  = \frac{{{x^{\alpha  + 1}}}}{{\alpha  + 1}} + C\). • \(\int {\frac{1}{x}dx}  = \ln \left| x \right| + C\).

Lời giải chi tiết

\(\begin{array}{l}f\left( 4 \right) - f\left( 1 \right) = \int\limits_1^4 {f'\left( x \right)dx}  = \int\limits_1^4 {\frac{{\sqrt x  - 1}}{x}dx}  = \int\limits_1^4 {\left( {{x^{ - \frac{1}{2}}} - \frac{1}{x}} \right)dx}  = \left. {\left( {2{x^{\frac{1}{2}}} - \ln {\rm{x}}} \right)} \right|_1^4\\ = \left( {{{2.4}^{\frac{1}{2}}} - \ln 4} \right) - \left( {{{2.1}^{\frac{1}{2}}} - \ln 1} \right) = 2 - 2\ln 2\end{array}\)

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