Bài 1. Tìm nguyên hàm \int 2^x\left(3^x+\dfrac{2^{-x}}{\sqrt{x+1}} \right) dx
Ta có: \displaystyle \int 2^x \left(3^x + \dfrac{2^{-x}}{\sqrt{x+1}} \right)\mbox{d}x = \displaystyle \int \left(6^x + \dfrac{1}{\sqrt{x+1}} \right)\mbox{x} = \displaystyle \int 6^x \mbox{d}x + \displaystyle \int \dfrac{\mbox{d}(x+1)}{(x+1)^{\frac{1}{2}}}
=\dfrac{6^x}{\ln 6}+ 2\sqrt{x+1} +C
Bài 2. Tính nguyên hàm \displaystyle \int \dfrac{x^2+x+1}{x^3-3x+2}dx
\displaystyle \int \dfrac{x^2+x+1}{x^3-3x+2}dx = \frac{1}{3}\int\frac{1}{x+2}dx+\frac{2}{3}\int \frac{1}{x-1}dx+\int \frac{1}{(x-1)^2}dx
=\frac{1}{3} \ln \left | x+2 \right |+\frac{2}{3} \ln \left | x-1 \right |- \frac{1}{x-1} + C
Bài 3. Tính \displaystyle \int xe^x \cos x dx.
I=x \cos x.e^x-\displaystyle \int \cos xd(e^x)+ \displaystyle \int x \sin xd(e^x)=x \cos x.e^x- \cos x.e^x+\displaystyle \int \sin x.e^xdx+ x \sin x.e^x-\displaystyle \int (\sin x+x \cos x)e^xdx
=x \cos x.e^x+x \sin x.e^x- \cos x.e^x-\displaystyle \int x \cos x.e^xdx
Suy ra 2I=x \cos x.e^x+x\sin x.e^x- \cos x.e^x\Rightarrow I=\dfrac{1}{2}[x \cos x+x \sin x- \cos x].e^x+C
Bài 4. Tìm nguyên hàm: \displaystyle \int x^3 \ln (2x)dx
Sau khi áp dụng nguyên hàm từng phần, ta được:
\displaystyle \int x^3 \ln (2x)\mbox{d}x=\dfrac{x^4}{4}\ln 2x - \displaystyle \int \dfrac{x^3}{4}\mbox{d}{x}= \dfrac{x^4}{4}\ln 2x - \dfrac{x^4}{16} +C
Bài 5. Tìm nguyên hàm : I= \displaystyle \int (x^2+1).e^{-x}\mbox{d}x
Xét hàm số: f(x) = (-x^2-2x-3)e^{-x}.
Ta có: f'(x)=(-2x-2)e^{-x}+(x^2+2x+3)e^{-x}=(x^2+1)e^{-x} nên I = f(x) +C = (-x^2-2x-3)e^{-x}+C
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