Question

निम्नलिखित फलन का समाकलन ज्ञात कीजिए-

${\displaystyle \frac{\cos 2x+2{\sin }^{2}x}{{\cos }^{2}x}}$

Answer 1

माना ${\displaystyle I=\int \frac{\cos 2x+2{\sin }^2}{{\cos }^{2}x}dx}$

${\displaystyle =\int \frac{({\cos }^{2}x-{\sin }^{2}x)+2{\sin }^{2}x}{{\cos }^{2}x}}$     .... [∵ cos 2x = cos² x - sin² x]

${\displaystyle =\int \frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}dx}$

${\displaystyle =\frac{1}{{\cos }^{2}x}dx}$

${\displaystyle =\int {\sec }^{2}xdx =\tan x+C}$