Giải thích các bước giải:
$K=\int\dfrac{dx}{x\sqrt{x^3+1}}$
Đặt $u=x^3$
$\to K=\int\dfrac{du}{3u\sqrt{u+1}}$
Đặt $v=\sqrt{u+1}$
$\to K=\int\dfrac{1}{3}.\dfrac{2}{v^2-1}dv$
$\to K=\int\dfrac{1}{3}.\dfrac{2}{(v-1)(v+1)}dv$
$\to K=\dfrac 13\int\dfrac{v+1-(v-1)}{(v-1)(v+1)}dv$
$\to K=\dfrac 13\int\dfrac{1}{v-1}-\dfrac{1}{v+1}dv$
$\to K=\dfrac 13(\ln|v-1|-\ln|v+1|)$
$\to K=-\dfrac{1}{3}\left(\ln \left|\sqrt{x^3+1}+1\right|-\ln \left|\sqrt{x^3+1}-1\right|\right)$
$\to I=-\dfrac{\ln \left(2\right)-\ln \left(\sqrt{2}+1\right)+\ln \left(\sqrt{2}-1\right)}{3}$
$\to B$
