Question

интеграл x корень x^2-5 dx

Answer 1

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

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