Question

Помогите решить неопределенный интеграл
$\int\limits (5-3x)^2^1 \, dx$


Answer 1

$\int (5-3x)^{21}dx=[\, t=5-3x\; ,\; dt=-3\, dx\; ,\; dx=-\frac{dt}{3}\, ]=\\\\=-\frac{1}{3}\int t^{21}dt=-\frac{1}{3}\cdot \frac{t^{22}}{22}+C=-\frac{1}{66}\cdot (5-3x)^{22} +C$

Answer 2

$\int\limits {(5-3x)^{21}} \, dx = \int\limits {(5-3x)^{21}} \, d(\frac{-3x}{-3})=\\ \\ =-\frac{1}{3}\int\limits {(5-3x)^{21}} \, d(-3x)=-\frac{1}{3}\int\limits {(5-3x)^{21}} \, d(5-3x)=\\\\ =-\frac{1}{3}*\frac{(5-3x)^{21+1}}{21+1}+C=-\frac{(5-3x)^{22}}{66}+C=-\frac{(3x-5)^{22}}{66}+C$