Question

интегралы номер 51 и 61

Answer 1

$\displaystyle\int tg^mnx\frac{dx}{cos^2nx}=\frac{1}{n}\int t^mdt=\frac{t^{m+1}}{n(m+1)}+C=\frac{tg^{m+1}(nx)}{n(m+1)}+C\\t=tgnx;dt=\frac{ndx}{cos^2nx};dx=\frac{cos^2nx}{n}dt$

$\displaystyle\int\limits^3_0e^\frac{x}{3}dx=3\int\limits^1_0 e^{t}dt=3e^t|^1_0=3e-3\approx5,155\\t=\frac{x}{3};x=3t;dx=3dt\\t_1=1;t_2=0$