Biết \(\displaystyle\int f(u)\mathrm{\,d}u=F(u)+C\). Mệnh đề nào dưới đây đúng?
\(\displaystyle\int f(2x-1)\mathrm{\,d}x=2F(2x-1)+C\) | |
\(\displaystyle\int f(2x-1)\mathrm{\,d}x=2F(x)-1+C\) | |
\(\displaystyle\int f(2x-1)\mathrm{\,d}x=\dfrac{1}{2}F(2x-1)+C\) | |
\(\displaystyle\int f(2x-1)\mathrm{\,d}x=F(2x-1)+C\) |
Chọn phương án C.
Đặt \(u=2x-1\Rightarrow \mathrm{\,d}u=2\mathrm{\,d}x\).
Khi đó $$\begin{aligned}
\displaystyle\int f(2x-1)\mathrm{\,d}x&=\displaystyle\int f(u)\cdot\dfrac{1}{2}\mathrm{\,d}u\\
&=\dfrac{1}{2}\displaystyle\int f(u)\mathrm{\,d}u\\
&=\dfrac{1}{2}F(u)+C.
&=\dfrac{1}{2}F(2x-1)+C.
\end{aligned}$$