# Find the integral of sinx/x.

$frac{d}{dx}frac{f(x)}{g(x)} = frac{f'(x)g(x)-g'(x)f(x)}{g^2(x)} frac{f(x)}{g(x)}=int frac{f'(x)g(x)}{g^2(x)},dx-int frac{g'(x)f(x)}{g^2(x)},dx int frac{g'(x)f(x)}{g^2(x)},dx=int frac{f'(x)}{g(x)},dx-frac{f(x)}{g(x)},dx int frac{sin(x)}{x},dx=int frac{xsin(x)}{x^2},dx=frac{1}{2}int frac{2xsin(x)}{x^2},dx g(x)=x^2$ $f(x)=sin(x) g'(x)=2x f'(x)=cos(x) int frac{g'(x)f(x)}{g^2(x)},dx=int frac{f'(x)}{g(x)},dx-frac{f(x)}{g(x)},dx frac{1}{2}int frac{2xsin(x)}{x^2},dx=frac{1}{2}int frac{cos(x)}{x},dx-frac{sin(x)}{x},dx int frac{cos(x)}{x},dx=frac{1}{2}int frac{2xcos(x)}{x^2},dx frac{1}{2}int frac{2xcos(x)}{x^2},dx=frac{1}{2}int frac{-sin(x)}{x},dx-frac{1}{2}frac{cos(x)}{x},dx$ $frac{1}{2}int frac{2xcos(x)}{x^2},dx=frac{1}{2}int frac{-sin(x)}{x},dx-frac{1}{2}frac{cos(x)}{x},dx int frac{sin(x)}{x},dx=frac{1}{2}left(frac{1}{2} int frac{-sin(x)}{x} ,dx – frac{1}{2}frac{cos(x)}{x}right)-frac{1}{2}frac{sin(x)}{x} frac{5}{4}int frac{sin(x)}{x},dx=-frac{1}{4}frac{cos(x)}{x}-frac{1}{2}frac{sin(x)}{x} int frac{sin(x)}{x},dx=-frac{1}{5}frac{cos(x)}{x}-frac{2}{5}frac{sin(x)}{x}$