Tính nguyên hàm

1.  $\int\limits x^2sin2xdx$

2.  $\int\limits x^2lnxdx$
$I =\int x^2 \ln x dx$ đặt $\ln x = u \Rightarrow \dfrac{1}{x}dx = du $ và $x^2 dx = dv \Rightarrow \dfrac{1}{3}x^3 =v$

$ = \dfrac{1}{3}x^3 \ln x - \dfrac{1}{3}\int \dfrac{1}{x} x^3dx = \dfrac{1}{3}x^3 \ln x - \dfrac{1}{3}\int x^2 dx = \dfrac{1}{3}x^3 \ln x - \dfrac{1}{3.3}x^3 +C$
Từng phần nào

$x^2 = u \Rightarrow 2xdx = du$ và $\sin 2x dx = dv \Rightarrow -\dfrac{1}{2}\cos 2x = v$

$I = -\dfrac{1}{2}x^2. \cos 2x +\dfrac{1}{2}\int x \cos 2x dx = -\dfrac{1}{2}x^2. \cos 2x +\dfrac{1}{2}I_1$

từng phần tiếp $I_1=\int x \cos 2x dx$ đặt $x = u \Rightarrow dx = du$ và $\cos 2x dx = dv \Rightarrow \dfrac{1}{2}\sin 2x = v$

$I_1 = \dfrac{1}{2}x \sin 2x - \dfrac{1}{2}\int \sin 2x dx = \dfrac{1}{2}x \sin 2x +\dfrac{1}{2}\cos 2x$

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