32.
Đặt \(\sqrt{1+lnx}=u\Rightarrow lnx=u^2-1\Rightarrow\dfrac{dx}{x}=2udu\)
\(\left\{{}\begin{matrix}x=1\Rightarrow u=1\\x=e\Rightarrow u=\sqrt{2}\end{matrix}\right.\)
\(I=\int\limits^{\sqrt{2}}_1\dfrac{\left(u^2-1\right).2udu}{u}=\int\limits^{\sqrt{2}}_1\left(2u^2-2\right)du=\left(\dfrac{2}{3}u^3-2u\right)|^{\sqrt{2}}_1=-\dfrac{2\sqrt{2}}{3}+\dfrac{4}{3}\)
\(\Rightarrow\left\{{}\begin{matrix}a=\dfrac{4}{3}\\b=-\dfrac{2}{3}\end{matrix}\right.\) \(\Rightarrow a+b=\dfrac{2}{3}\)
33.
Đặt \(\left\{{}\begin{matrix}u=x\\dv=\left(x-1\right)^{1000}dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=dx\\v=\dfrac{\left(x-1\right)^{1001}}{1001}\end{matrix}\right.\)
\(\Rightarrow I=\dfrac{x\left(x-1\right)^{1001}}{1001}|^3_1-\dfrac{1}{1001}\int\limits^3_1\left(x-1\right)^{1001}dx=\dfrac{3.2^{1001}}{1001}-\dfrac{1}{1001.1002}.\left(x-1\right)^{1002}|^3_1\)
\(=\dfrac{3.2^{1001}}{1001}-\dfrac{2^{1002}}{1001.1002}=\dfrac{2^{1001}}{1001}\left(3-\dfrac{2}{1002}\right)=\dfrac{1502.2^{1001}}{501501}\)
