a) $\displaystyle\int x^3\sqrt{x^2 +1}dx$
Đặt $u = x^2 +1$
$\to du =2xdx$
Ta được:
$\dfrac12\displaystyle\int (u-1)\sqrt udu$
$= \dfrac12\displaystyle\int u^{\tfrac32} -\dfrac12\displaystyle\int \sqrt udu$
$=\dfrac12\cdot\dfrac{2u^{\tfrac52}}{5} -\dfrac12\cdot \dfrac{2u^{\tfrac32}}{3} + C$
$=\dfrac{(x^2+1)^{\tfrac52}}{5} - \dfrac{(x^2+1)^{\tfrac32}}{3} +C$
$=\dfrac{1}{15}\sqrt{(x^2+1)^3}(3x^2 -2) + C$
b) $\displaystyle\int \dfrac{e^{\tan x}}{\cos^2x}dx$
Đặt $t = \tan x$
$\to dt =\dfrac{1}{\cos^2x}dx$
Ta được:
$\displaystyle\int e^tdt$
$= e^t + C$
$= e^{\tan x} + C$
