Question

cho \(\int f\left(4x\right)dx\) = x²+3x+C. Mệnh đề nào sau đây đúng?

A. \(\int f\left(x+2\right)dx\) =x²+7x+C

B.\(\int f\left(x+2\right)dx\) =\(\frac{x^2}{2}\)+4x+C

C.\(\int f\left(x+2\right)dx\)=\(\frac{x^2}{4}\)+2x+C

D.\(\int f\left(x+2\right)dx\)=\(\frac{x^2}{4}\)+4x+C

Giúp mình bài này với, cám ơn mọi người nhiều

Answer 1

\(\int f\left(4x\right)dx=\frac{1}{4}\int f\left(4x\right)d\left(4x\right)=\frac{1}{16}\left(4x\right)^2+\frac{3}{4}\left(4x\right)+C\)

\(\Rightarrow\int f\left(4x\right)d\left(4x\right)=\frac{1}{4}\left(4x\right)^2+3.\left(4x\right)+C\)

\(\Rightarrow\int f\left(x+2\right)dx=\int f\left(x+2\right)d\left(x+2\right)=\frac{1}{4}\left(x+2\right)^2+3\left(x+2\right)+C\)

\(=\frac{1}{4}x^2+4x+C\)