Cho hàm số (fleft( x right)) thỏa mãn ({left( {fleft( x right)} right)^2} + fleft( x right)

Ta có:

(begin{array}{l}left[ {f’left( x right).fleft( x right)} right]’\ = f”left( x right).fleft( x right) + f’left( x right).f’left( x right)\ = {left( {f’left( x right)} right)^2} + fleft( x right).f”left( x right)end{array})

Do đó: (left[ {f’left( x right).fleft( x right)} right]’ = 15{x^4} + 12x,,,forall x in mathbb{R}).

Lấy nguyên hàm hai vế ta được:

(begin{array}{l}int {left[ {f’left( x right).fleft( x right)} right]’dx}  = int {left( {15{x^4} + 12x} right)dx} \ Leftrightarrow f’left( x right).fleft( x right) = 3{x^5} + 6{x^2} + Cend{array})

Thay (x = 0) ta có: (f’left( 0 right).fleft( 0 right) = C Leftrightarrow C = 1).

( Rightarrow f’left( x right).fleft( x right) = 3{x^5} + 6{x^2} + 1)

Tiếp tục lấy nguyên hàm hai vế ta được:

(begin{array}{l}int {f’left( x right)fleft( x right)dx}  = int {left( {3{x^5} + 6{x^2} + 1} right)dx} \ Leftrightarrow frac{{{f^2}left( x right)}}{2} = frac{1}{2}{x^6} + 2{x^3} + x + C’end{array})

Thay (x = 0) ta có: (frac{{{f^2}left( 0 right)}}{2} = C’ Leftrightarrow C’ = frac{1}{2}).

(begin{array}{l} Rightarrow frac{{{f^2}left( x right)}}{2} = frac{1}{2}{x^6} + 2{x^3} + x + frac{1}{2}\ Leftrightarrow {f^2}left( x right) = {x^6} + 4{x^3} + 2x + 1end{array})

Vậy ({f^2}left( 1 right) = 1 + 4 + 2 + 1 = 8).