ГДЗ по алгебре и начала математического анализа 10 класс Алимов Задание 998

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\[1)\ f(x) = \frac{x}{x - 3} = \frac{x - 3 + 3}{x - 3} =\]

\[= 1 + \frac{3}{x - 3};\]

\[F(x) = 1 \bullet \frac{x^{1}}{1} + 3 \bullet \frac{1}{1} \bullet \ln(x - 3) =\]

\[= x + 3\ln(x - 3) + C.\]

\[2)\ f(x) = \frac{x - 1}{x^{2} + x - 2} =\]

\[= \frac{x - 1}{x^{2} - x + 2x - 2} =\]

\[= \frac{x - 1}{x(x - 1) + 2(x - 1)} = \frac{1}{x + 2};\]

\[F(x) = \frac{1}{1} \bullet \ln(x + 2) =\]

\[= \ln(x + 2) + C.\]

\[3)\ f(x) = \cos^{2}x = \frac{1 + \cos{2x}}{2};\]

\[F(x) = \frac{1}{2} \bullet \frac{x^{1}}{1} + \frac{1}{2} \bullet \frac{1}{2}\sin{2x} =\]

\[= \frac{x}{2} + \frac{1}{4}\sin{2x} = \frac{2x + \sin{2x}}{4} + C.\]

\[4)\ f(x) = \sin{3x} \bullet \cos{5x};\]

\[= \frac{4\cos{2x} - \cos{8x}}{16} + C.\]