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Câu 1
\(F\left(x\right)=\int\left(2x^2+6x\right)e^{2x}dx\)
Đặt \(\left\{{}\begin{matrix}u=2x^2+6x\\dv=e^{2x}dx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=\left(4x+6\right)dx\\v=\dfrac{1}{2}e^{2x}\end{matrix}\right.\)
\(\Rightarrow F\left(x\right)=\dfrac{1}{2}e^{2x}\left(2x^2+6x\right)-\int\dfrac{1}{2}e^{2x}\left(4x+6\right)dx\)
\(=e^{2x}\left(x^2+3x\right)-\int\left(2x+3\right)e^{2x}dx\)
Gọi \(I=\int\left(2x+3\right)e^{2x}dx\)
Đặt \(\left\{{}\begin{matrix}u=2x+3\\dv=e^{2x}dx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=2dx\\v=\dfrac{1}{2}e^{2x}\end{matrix}\right.\)
\(\Rightarrow I=\dfrac{1}{2}e^{2x}\left(2x+3\right)-\int\dfrac{1}{2}e^{2x}2dx=\dfrac{1}{2}e^{2x}\left(2x+3\right)-\dfrac{1}{2}e^{2x}\)
