Bài tập 2 trang 141 SGK Toán 12 NC

a)

\(\begin{array}{l}
\int {\left( {\sqrt x  + \sqrt[3]{x}} \right)dx}  = \int {\left( {{x^{\frac{1}{2}}} + {x^{\frac{1}{3}}}} \right)} dx\\
 = \frac{{{x^{\frac{3}{2}}}}}{{\frac{3}{2}}} + \frac{{{x^{\frac{4}{2}}}}}{{\frac{4}{2}}} + C = \frac{2}{3}{x^{\frac{3}{2}}} + \frac{3}{4}{x^{\frac{4}{2}}} + C
\end{array}\)

b)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
\int {\frac{{x\sqrt x  + \sqrt x }}{{{x^2}}}} dx\\
 = \int {\frac{1}{{\sqrt x }}dx}  + \int {\frac{{dx}}{{x\sqrt x }}} 
\end{array}\\
\begin{array}{l}
 = \int {{x^{ - \frac{1}{2}}}dx}  + \int {{x^{ - \frac{3}{2}}}dx} \\
 = \frac{{{x^{\frac{1}{2}}}}}{{\frac{1}{2}}} + \frac{{{x^{ - \frac{1}{2}}}}}{{ - \frac{1}{2}}} + C\\
 = 2\sqrt x  - \frac{2}{{\sqrt x }} + C
\end{array}
\end{array}\)

c)

\(\begin{array}{*{20}{l}}
{\int 4 {{\sin }^2}xdx = \int 2 \left( {1 - \cos 2x} \right)dx}\\
\begin{array}{l}
 = 2\int {dx}  - 2\int {\cos 2x} dx\\
 = 2x - \sin 2x + C
\end{array}
\end{array}\)

d) \(\int {\frac{{1 + \cos 4x}}{2}dx}  = \frac{x}{2} + \frac{1}{8}\sin 4x + C\)

-- Mod Toán 12 HỌC247