Question

1)int sin^3 5xdx

2)int 5sqrtx -2x^3 +4/x^2 .dx

3)int 2x+3/sqrt 2x^2-x+6

Answer 1

$1); ; int sin^35x, dx=int sin^25xcdot sin5x, dx=int (1-cos^25x)cdot sin5x, dx=\\=int sin5x, dx-int cos^25xcdot underbrace {sin5x, dx}_{-1/5cdot d(cos5x)}=-frac{1}{5}cos5x+ frac{1}{5}cdot frac{cos^35x}{3}+C; ;\\\2); ; int frac{5sqrt{x}-2x^3+4}{x^2}, dx=int (5x^{-3/2}-2x+4x^{-2})dx=\\=frac{5x^{-1/2}}{-1/2}-frac{2x^2}{2}+frac{4x^{-1}}{-1}+C=-frac{10}{sqrt{x}}-x^2-frac{4}{x}+C; ;$