Question

интеграл x корень x^2-5 dx

Answer 1

$int xsqrt{x^2-5}, dx=[, x=frac{sqrt5}{cost},; dx=frac{sqrt5sint}{cos^2t}dt, ]=\\=int frac{sqrt5}{cost}cdot sqrt{frac{5}{cos^2t}-5}}cdot frac{sqrt5sint}{cos^2t}, dt=int frac{sqrt5}{cost}cdot sqrt{5tg^2t}cdot frac{sqrt5sint}{cos^2t}, dt=\\=5sqrt5cdot int frac{tg^2tcdot sint}{cos^3t} dt=5sqrt5cdot int frac{sin^3t}{cos^5t} dt=\\=5sqrt5int tg^3tcdot frac{dt}{cos^2t}=5sqrt5int tg^3tcdot d(tgt)=5sqrt5cdot frac{tg^4t}{4}+C=$

$=frac{5sqrt5}{4}cdot tg^4(arccosfrac{sqrt5}{x})+C$