Giải thích các bước giải:
1.Ta có:
$\displaystyle\int\dfrac{(x^2+2)^2}{x^2}dx$
$=\displaystyle\int\dfrac{x^4+4x^2+4}{x^2}dx$
$=\displaystyle\int x^2+4+\dfrac{4}{x^2}dx$
$=\dfrac13x^3+4x-\dfrac4x+C$
2.Ta có:
$\displaystyle\int (\sqrt[3]{x}+x\sqrt{x})dx$
$=\displaystyle\int (x^{\dfrac13}+x^{\dfrac32})dx$
$=\dfrac{3}{4}x^{\dfrac{4}{3}}+\dfrac{2}{5}x^{\dfrac{5}{2}}+C$
3.Ta có:
$\displaystyle\int (\sqrt{x}+1)^2dx$
$=\displaystyle\int (x+2\sqrt{x}+1)dx$
$=\displaystyle\int (x+2\cdot x^{\dfrac12}+1)dx$
$=\dfrac{x^2}{2}+\dfrac{4}{3}x^{\dfrac{3}{2}}+x+C$
4.Ta có:
$\displaystyle\int \dfrac{x\sqrt[4]{x}+\sqrt[3]{x^2}}{\sqrt{x}}dx$
$=\displaystyle\int \dfrac{x^{\dfrac54}+x^{\dfrac23}}{x^{\dfrac12}}dx$
$=\displaystyle\int x^{\dfrac54-\dfrac12}+x^{\dfrac23-\dfrac12}dx$
$=\displaystyle\int x^{\dfrac34}+x^{\dfrac16}dx$
$=\dfrac{4}{7}x^{\dfrac{7}{4}}+\dfrac{6}{7}x^{\dfrac{7}{6}}+C$
