Question

найти первообразную функцию f(x)=1-sin5х

Answer 1

$F(x)= \int\limits {[1-sin(5x)}] \, dx = \int\limits {1} \, dx - \int\limits {sin(5x)} \, dx =\\\\ =x- \int\limits {sin(5x)} \, d(\frac{5x}{5})=x-\frac{1}{5} \int\limits {sin(5x)} \, d(5x)=\\\\ =x-\frac{1}{5} \int\limits {sin(t)} \, dt =x-\frac{1}{5}*[-cos(t)]+C=x+\frac{1}{5}cos(t)+C=\\\\ =x+\frac{cos(5x)}{5}+C$