Решить уравнение: 1) 2^(x-5) = 32^(x-3);

1) $2^{x-5} = 32^{x-3}$ $2^{x-5} = (2^5)^{x-3}$ $2^{x-5} = 2^{5(x-3)}$ $x - 5 = 5x - 15$ $10 = 4x$ $x = 2.5$ 2) $\log_4(3x + 7) = 2$ $3x + 7 = 4^2$ $3x + 7 = 16$ $3x = 9$ $x = 3$ 3) $9^{6+x} \le 81^{3-7x}$ $(3^2)^{6+x} \le (3^4)^{3-7x}$ $3^{12+2x} \le 3^{12-28x}$ $12 + 2x \le 12 - 28x$ $30x \le 0$ $x \le 0$ 4) $y = 6x^8 + 3x^3 - 6$ $y' = (6x^8)' + (3x^3)' - (6)'$ $y' = 48x^7 + 9x^2$ 5) $y = x^4 \cdot e^x$ $y' = (x^4)' \cdot e^x + x^4 \cdot (e^x)'$ $y' = 4x^3 \cdot e^x + x^4 \cdot e^x = x^3 e^x (4 + x)$ 6) $y = \frac{2x^6 - 7x^2}{10x - 3}$ $y' = \frac{(2x^6 - 7x^2)'(10x - 3) - (2x^6 - 7x^2)(10x - 3)'}{(10x - 3)^2}$ $y' = \frac{(12x^5 - 14x)(10x - 3) - (2x^6 - 7x^2)(10)}{(10x - 3)^2}$ $y' = \frac{120x^6 - 36x^5 - 140x^2 + 42x - 20x^6 + 70x^2}{(10x - 3)^2}$ $y' = \frac{100x^6 - 36x^5 - 70x^2 + 42x}{(10x - 3)^2}$ 7) $s = \frac{4}{3}t^3 + 3t^2 - 8t + 7$, $t=4c$ $v(t) = s'(t) = 4t^2 + 6t - 8$ $v(4) = 4(4^2) + 6(4) - 8 = 64 + 24 - 8 = 80$ $a(t) = v'(t) = 8t + 6$ $a(4) = 8(4) + 6 = 32 + 6 = 38$ 8) $\int(3x^2 + 7 \sin x) dx = x^3 - 7 \cos x + C$