Đáp án:
\(\begin{array}{l}
a)\,\,\frac{3}{2}\sin 2x + \frac{7}{5}\cos 5x + C\\
b)\,\,\frac{x}{2} - \frac{{\sin 2x}}{4} + C\\
c)\,\,\frac{{{x^4}}}{4} + \frac{{2{x^3}}}{3} + \frac{{{x^2}}}{2} + C
\end{array}\)
Giải thích các bước giải:
\(\begin{array}{l}
a)\,\,\int {\left( {3\cos 2x - 7\sin 5x} \right)dx} \\
= 3\int {\cos 2xdx} - 7\int {\sin 5xdx} \\
= \frac{3}{2}\sin 2x + \frac{7}{5}\cos 5x + C\\
b)\,\,\int {{{\sin }^2}xdx} \\
= \int {\frac{{1 - \cos 2x}}{2}dx} \\
= \frac{1}{2}\int {dx} - \frac{1}{2}\int {\cos 2xdx} \\
= \frac{x}{2} - \frac{1}{2}.\frac{{\sin 2x}}{2} + C\\
= \frac{x}{2} - \frac{{\sin 2x}}{4} + C\\
c)\,\,\int {x{{\left( {x + 1} \right)}^2}dx} \\
= \int {x\left( {{x^2} + 2x + 1} \right)dx} \\
= \int {\left( {{x^3} + 2{x^2} + x} \right)dx} \\
= \frac{{{x^4}}}{4} + \frac{{2{x^3}}}{3} + \frac{{{x^2}}}{2} + C
\end{array}\)
