Nguyên hàm [Lần 2]

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 $I = f(x) +C = (-x^2-2x-3)e^{-x}+C$