Tìm nguyên hàm của các hàm số sau :
\(\begin{array}{l}
a)f\left( x \right) = 3{x^2} + \frac{x}{2}\\
b)f(x) = 2{x^3} - 5x + 7\\
c)f\left( x \right) = \frac{1}{{{x^2}}} - {x^2} - \frac{1}{3}\\
d)f(x) = {x^{ - \frac{1}{3}}}\\
e)f(x) = {10^{2x}}
\end{array}\)
a)
\(\begin{array}{l}
\int {\left( {3{x^2} + \frac{x}{2}} \right)} dx\\
= 3\int {{x^2}dx} + \frac{1}{2}\int {xdx} \\
= {x^3} + \frac{{{x^2}}}{4} + C
\end{array}\)
b)
\(\begin{array}{l}
\int {\left( {2{x^3} - 5x + 7} \right)} dx\\
= 2\int {{x^3}dx} - 5\int {xdx} + 7\int {dx} \\
= \frac{{{x^4}}}{2} - \frac{{5{x^2}}}{4} + 7x + C
\end{array}\)
c)
\(\begin{array}{l}
\int {\left( {\frac{1}{{{x^2}}} - {x^2} - \frac{1}{3}} \right)} dx\\
= \int {{x^{ - 2}}dx} - \int {{x^2}dx} - \frac{1}{3}\int {dx} \\
= - \frac{1}{x} - \frac{{{x^3}}}{3} - \frac{x}{3} + C
\end{array}\)
d) \(\int {\left( {{x^{ - \frac{1}{3}}}} \right)} dx = \frac{{{x^{\frac{2}{3}}}}}{{\frac{2}{3}}} + C = \frac{3}{2}{x^{\frac{2}{3}}} + C\)
e) \(\int {{{10}^{2x}}} dx = \frac{{{{10}^{2x}}}}{{2.\ln 10}} + C\)
-- Mod Toán 12
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