Question

Найти неопределенные интегралы

Answer 1

$1.6. \int\limits {\frac{1}{x^2+4x+5} } \, dx = \int\limits {\frac{1}{x^2+4x+2+5-2} } \, dx = \int\limits {\frac{1}{(x+\sqrt{2})^2+3 } } \, dx = \frac{1}{\sqrt{3} } arctg\frac{x+2}{\sqrt{3} } + c\\1.7. \int\limits {xe^x} \, dx = |u=x, dv = e^xdx, du = dx, v = e^x| = xe^x - \int\limits {e^x} \, dx = xe^x - e^x + c\\ = e^x(x-1)+c\\1.8. \int\limits {\frac{1}{x^2\sqrt{x^2+1} } } \, dx = |x=tgt, t=arctgx, dx=\frac{1}{cos^2t}dt| = \int\limits {\frac{1}{cos^2t*tg^2t\sqrt{tg^2t+1} } } \, dx =$$\int\limits {\frac{1}{cost*tg^2t} } \, dt = \int\limits {\frac{1}{sint} } \, dt = ln|sint| + c = ln|arctx| + c$