Tìm
\(\begin{array}{*{20}{l}}
{a)\int {\left( {\sqrt x + \sqrt[3]{x}} \right)dx} }\\
{b)\int {\frac{{x\sqrt x + \sqrt x }}{{{x^2}}}} dx}\\
{c)\int {4{{\sin }^2}xdx} }\\
{d)\int {\frac{{1 + \cos 4x}}{2}dx} }
\end{array}\)
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
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