17) ∫ 1 + t a n 2 x d x {\displaystyle \int _{}^{}1+tan^{2}xdx} =
∫ 1 + s i n 2 x c o s 2 x d x {\displaystyle \int _{}^{}1+{\frac {sin^{2}x}{cos^{2}x}}dx} =
∫ c o s 2 x + s i n 2 x c o s 2 x d x {\displaystyle \int _{}^{}{\frac {cos^{2}x+sin^{2}x}{cos^{2}x}}dx}
cos 2 x + s i n 2 x = 1 {\displaystyle \cos ^{2}x+sin^{2}x=1} thus,
∫ 1 c o s 2 x d x {\displaystyle \int _{}^{}{\frac {1}{cos^{2}x}}dx} =
∫ sec 2 x d x {\displaystyle \int _{}^{}\sec ^{2}xdx} =
t a n x + C {\displaystyle tanx+C}
1 3 5 7 8 9 10 11 13 15 17 19 20 21 23 25 27 28 29 31 33 35 37 39 41 53
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thus,
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