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Giải bài tập 4.5 trang 11 SGK Toán 12 tập 2 - Kết nối tri thức

Cho hàm số \(y = f\left( x \right)\) xác định trên khoảng \(\left( {0; + \infty } \right)\). Biết rằng \(f'\left( x \right) = 2x + \frac{1}{{{x^2}}}\) với mọi \(x \in \left( {0; + \infty } \right)\) và \(f\left( 1 \right) = 1\). Tính giá trị f(4).

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Đề bài

 
 
Cho hàm số \(y = f\left( x \right)\) xác định trên khoảng \(\left( {0; + \infty } \right)\). Biết rằng \(f'\left( x \right) = 2x + \frac{1}{{{x^2}}}\) với mọi \(x \in \left( {0; + \infty } \right)\) và \(f\left( 1 \right) = 1\). Tính giá trị f(4).
 

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về khái niệm nguyên hàm của một hàm số để giải: Cho hàm số f(x) xác định trên một khoảng K (hoặc một đoạn, hoặc một nửa khoảng). Hàm số F(x) được gọi là một nguyên hàm của hàm số f(x) trên K nếu \(F'\left( x \right) = f\left( x \right)\) với mọi x thuộc K. Sử dụng kiến thức về nguyên hàm của hàm số lũy thừa để tính: \(\int {{x^\alpha }dx}  = \frac{{{x^{\alpha  + 1}}}}{{\alpha  + 1}} + C\left( {\alpha  \ne  - 1} \right)\)
 

Lời giải chi tiết

Vì \(f'\left( x \right) = 2x + \frac{1}{{{x^2}}}\) nên \(f\left( x \right) = \int {f'\left( x \right)dx}  = \int {\left( {2x + \frac{1}{{{x^2}}}} \right)dx}  = 2\int {xdx}  + \int {{x^{ - 2}}dx}  = {x^2} - \frac{1}{x} + C\) Mà \(f\left( 1 \right) = 1\) nên \(1 - 1 + C = 1\), suy ra \(C = 1\). Do đó, hàm số \(f\left( x \right) = {x^2} - \frac{1}{x} + 1\) Vậy \(f\left( 4 \right) = {4^2} - \frac{1}{4} + 1 = \frac{{67}}{4}\)
 

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