Question 1
Durée : 15 mn
Note maximale : 5
Question
Calculer la primitive \(I_1=\int\frac{dx}{x+\sqrt[3]{x}}.\)
Poser\( x = t^3.\)
Solution
Posons\( x = t^3 \Leftrightarrow dx = 3 t^2 dt,\) et
\(I_1=\int\frac{3t^2dt}{t^3+t}~~\color{red}\text{ (2 pts)}\)
\(I_1=\frac32\int\frac{2t}{t^2+1}dt\)
\(=\frac32\ln(t^2+1)+C\)
\(\color{blue}I_1=\frac32\ln(\sqrt[3]{x^2}+1)+C~~\color{red}\text{ (3 pts)}\)