Cho F(x)= nguyên hàm (x+1)f′(x)dx. Tính I=∫f(x)dx theo F(x).

3/20

Cho \[F\left( x \right) = \smallint \left( {x + 1} \right)f'\left( x \right)dx\]. Tính \[I = \smallint f(x)dx\;\] theo F(x).

\[I = \left( {x + 1} \right)f\left( x \right) - 2F\left( x \right) + C\]

\[I = F\left( x \right) - \left( {x + 1} \right)f\left( x \right)\]

\[I = \left( {x + 1} \right)f\left( x \right) + C\]

\[I = \left( {x + 1} \right)f\left( x \right) - F\left( x \right) + C\]

Giải thích

Đặt\(\left\{ {\begin{array}{*{20}{c}}{u = x + 1}\\{dv = f\prime (x)dx}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}}{du = dx}\\{v = f(x)}\end{array}} \right.\)

\[ \Rightarrow F\left( x \right) = \left( {x + 1} \right)f\left( x \right) - \smallint f\left( x \right)dx + C\]

\[ \Rightarrow I = \smallint f\left( x \right)dx = \left( {x + 1} \right)f\left( x \right) - F\left( x \right) + C.\]

Đáp án cần chọn là: D