Question

Срочно 30 баллов
Нужно решить методом который указан

Answer 1

$1. \int e^{\cos(2x+4)}\sin(2x+4)\:dx =\\t = 2x+4, \;dt = 2\:dx\\= \frac12\cdot\int e^{\cos t}\sin t\:dt =\\u = \cos t, \;du = -\sin t\:dt\\= -\frac12\cdot\int e^u\:du = -\frac12\cdot e^u + C = -\frac12\cdot e^{cos(2x+4)} + C$

$2. \int x\sqrt[4]{2+x^2}\:dx =\\t = 2 +x^2, \;dt = 2x\:dx\\= \frac12\int\sqrt[4]{t}\:dt = \frac12\int t^{\frac14}\:dt = \frac12\cdot\frac45\cdot t^{\frac54} + C = \frac25\cdot\sqrt[4]{t^5} + C = \frac25\cdot\sqrt[4]{(2+x^2)^5} + C$

$3. \int\frac{\cos 3x\:dx}{\sqrt{5+4\sin 3x}} = \\\\t = 5 + 4\sin3x, \;dt = 12\cos3x\:dx\\= \frac1{12}\int\frac{dt}{\sqrt t} = \frac1{12}\int t^{-\frac12}\:dt = \frac16\cdot\sqrt t + C = \frac16\cdot\sqrt{5+4\sin 3x} + C$