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2.
\(I=\int e^{3x}.3^xdx\)
Đặt \(\left\{{}\begin{matrix}u=3^x\\dv=e^{3x}dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=3^xln3dx\\v=\dfrac{1}{3}e^{3x}\end{matrix}\right.\)
\(\Rightarrow I=\dfrac{1}{3}e^{3x}.3^x-\dfrac{ln3}{3}\int e^{3x}.3^xdx=\dfrac{1}{3}e^{3x}.3^x-\dfrac{ln3}{3}.I\)
\(\Rightarrow\left(1+\dfrac{ln3}{3}\right)I=\dfrac{1}{3}e^{3x}.3^x\)
\(\Rightarrow I=\dfrac{1}{3+ln3}.e^{3x}.3^x+C\)
1.
\(I=\int\left(2x-1\right)e^{\dfrac{1}{x}}dx=\int2x.e^{\dfrac{1}{x}}dx-\int e^{\dfrac{1}{x}}dx\)
Xét \(J=\int2x.e^{\dfrac{1}{x}}dx\)
Đặt \(\left\{{}\begin{matrix}u=e^{\dfrac{1}{x}}\\dv=2xdx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=-\dfrac{e^{\dfrac{1}{x}}}{x^2}dx\\v=x^2\end{matrix}\right.\)
\(\Rightarrow J=x^2.e^{\dfrac{1}{x}}+\int e^{\dfrac{1}{x}}dx\)
\(\Rightarrow I=x^2.e^{\dfrac{1}{x}}+C\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a/ \(\int\dfrac{x^2-3x+1}{x}dx=\int\left(x-3+\dfrac{1}{x}\right)dx=\int x.dx-3x+\int\dfrac{dx}{x}=\dfrac{1}{2}.x^2-3x+ln\left|x\right|+C\)
b/ \(I=\int x.e^{2x}dx\)
\(\left\{{}\begin{matrix}u=x\\dv=e^{2x}dx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=dx\\v=\dfrac{1}{2}e^{2x}\end{matrix}\right.\)
\(\Rightarrow I=\dfrac{1}{2}.x.e^{2x}-\dfrac{1}{2}\int e^{2x}.dx=\dfrac{1}{2}x.e^{2x}-\dfrac{1}{4}e^{2x}\)
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