Question

Интеграл от -1 до 0 (3/(5х-1)^3)dx

Answer 1

$\int\limits^{0}_{-1} {\frac{3}{(5x-1)^3}} \, dx = 3*\int\limits^{0}_{-1} {\frac{1}{(5x-1)^3}} \, d(\frac{5x}{5}) =\\\\ = 3*\frac{1}{5}*\int\limits^{0}_{-1} {(5x-1)^{-3}} \, d(5x) =\\\\ = \frac{3}{5}*\int\limits^{0}_{-1} {(5x-1)^{-3}} \, d(5x-1) =\\\\ = \frac{3}{5}*\int\limits^{5*0-1}_{5*(-1)-1} {t^{-3}} \, dt =\\\\ =\frac{3}{5}*\int\limits^{-1}_{-6} {t^{-3}} \, dt=\\\\ =\frac{3}{5}*\frac{t^{-3+1}}{-3+1}\ |^{-1}_{-6}=$

$=-\frac{3}{10}*\frac{1}{t^2}\ |^{-1}_{-6}=\\\\ =-\frac{3}{10}*[\frac{1}{(-1)^2}-\frac{1}{(-6)^2}]=\\\\ =-\frac{3}{10}*[1-\frac{1}{36}]=\\\\ =-\frac{3}{10}*\frac{35}{36}=\\\\ =-\frac{1}{2}*\frac{7}{12}=\\\\ =-\frac{7}{24}$