Question

Вычислить следующие неопределённые интегралы.

Answer 1

$1)\int\limits {x\sqrt{x+2}} \, dx = | x+2 = t^2 => dx = 2tdt| = \int\limits {(t^2-2)t*2t} \, dt =2\int\limits {(t^4 - 2t^2)} \, dt =2(\frac{1}{5}t^5 - \frac{2}{3}t^3) + c = \frac{2}{5}(x+2)^2\sqrt{x+2} - \frac{4}{3}(x+2)\sqrt{x+2} + c\\2)\int\limits {\frac{dx}{x^2+8x+7}} = \int\limits {\frac{dx}{(x+4)^2 - 9} = \int\limits {\frac{dx}{(x+7)(x+1)} = \frac{1}{6}\int\limits(\frac{1}{x+1}  - \frac{1}{x+7})dx =$$\frac{1}{6}(ln(x+1)-ln(x+7) + ln(c)) = \frac{1}{6}ln\frac{c(x+1)}{x+7}$

$3)\int {cos(5x)cos(x)} \, dx = \frac{1}{2}\int {cos(6x) + cos(4x)} \, dx = \frac{1}{2}(\frac{1}{6}sin(6x) + \frac{1}{4}sin(4x)) +c = \frac{1}{12}sin(6x) + \frac{1}{8}sin(4x) + c$