Tìm nguyên hàm của:
1. [tex]\int (\frac{1}{x}-\frac{1}{x^{2}}).e^{x} dx[/tex]
2. [tex]\int sinx.e^{2x} dx[/tex]
3. [tex]\int (x^{3}-3x^{2}+2).e^{x} dx[/tex]
4. [tex]\int lnx dx[/tex]
5. [tex]\int x.ln(x^{2}+16) dx[/tex]
6. [tex]\int ln(2x+3) dx[/tex]
1. $\int{(\dfrac{1}{x}-\dfrac{1}{x^2})e^xdx}=\int{\dfrac{xe^x-e^x}{x^2}dx}=\int{(\dfrac{e^x}{x})'dx}=\dfrac{e^x}{x}+C$
2. Đặt $u=e^{2x}\Rightarrow du=2e^{2x}dx$
$dv=\sin xdx\Rightarrow v=-\cos x$
$\int \sin x.e^{2x}dx=-e^{2x}\cos x+\int 2e^{2x}\cos xdx$
Đặt $u=e^{2x}\Rightarrow du=2e^{2x}dx$
$dw=\cos xdx\Rightarrow w=\sin x$
$\int 2e^{2x}\cos xdx=2e^{2x}\sin x-2\int2e^{2x}\sin xdx$
Suy ra $\int \sin x.e^{2x}dx=-e^{2x}\cos x+2e^{2x}\sin x-4\int e^{2x}\sin xdx$
$\Leftrightarrow 5\int \sin x.e^{2x}dx=-e^{2x}\cos x+2e^{2x}\sin x$
$\Rightarrow \int \sin x.e^{2x}dx=-\dfrac{1}{5}e^{2x}\cos x+\dfrac{2}{5}e^{2x}\sin x+C$
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https://diendan.hocmai.vn/threads/thong-bao-noi-quy-box-toan-hmf.837036/