Question

Integral (x square + 1) cot (x cube +3x)dx

Answer 1

`\bf \displaystyle \int~(x^2+1)cot(x^3+3x)\cdot dx\\\\ -------------------------------\\\\ \stackrel{substitution}{u=x^3+3x}\implies \cfrac{du}{dx}=3x^2+3\implies \cfrac{du}{dx}=3(x^2+1) \\\\\\ \cfrac{du}{3(x^2+1)}=dx\\\\ -------------------------------\\\\ \displaystyle \int~\underline{(x^2+1)} cot(u)\cdot \cfrac{du}{3\underline{(x^2+1)}}\implies \cfrac{1}{3}\int~cot(u)\cdot du \\\\\\ \cfrac{1}{3}( ln|sin(u)|)+C\implies \cfracsin(x^3+3x){3}+C`