17) ∫ 1 + t a n 2 x ∗ d x = d d x ( t a n ( x ) ) {\displaystyle \int {}{}1+tan^{2}x*dx={\frac {d}{dx}}(tan(x))}
d d x ( t a n ( x ) ) = d d x ( s i n ( x ) c o s ( x ) ) = c o s 2 x − ( − ( s i n ( x ) ) ( s i n ( x ) ) c o s 2 ( x ) {\displaystyle {\frac {d}{dx}}(tan(x))={\frac {d}{dx}}\left({\frac {sin(x)}{cos(x)}}\right)={\frac {cos^{2}x-(-(sin(x))(sin(x))}{cos^{2}(x)}}}
= c o s 2 x + s i n 2 ( x ) ) c o s 2 ( x ) = c o s 2 x c o s 2 ( x ) + s i n 2 x c o s 2 ( x ) {\displaystyle ={\frac {cos^{2}x+sin^{2}(x))}{cos^{2}(x)}}={\frac {cos^{2}x}{cos^{2}(x)}}+{\frac {sin^{2}x}{cos^{2}(x)}}}
= 1 + t a n 2 ( x ) {\displaystyle =1+tan^{2}(x)}
