Question

निम्नलिखित फलन का समाकलन कीजिए।

${\displaystyle \frac{2+\sin 2x}{1+\cos 2x}{e}^x}$

Answer 1

माना ${\displaystyle I=\int \frac{2+\sin 2x}{1+\cos 2x}{e}^x dx}$

${\displaystyle \int \frac{2+2\sin x\cos x}{1+2{\cos }^{2}x-1}{e}^x dx}$

${\displaystyle =\int \frac{2(1+\sin x\cos x)}{2{\cos }^{2}x}{e}^{x}dx}$

${\displaystyle =\int ({\sec }^{2}x\cdot e^x+\tan \cdot e^x) dx}$

${\displaystyle =\int e^x({\sec }^{2}x+\tan x)dx}$

e^x tan x = t प्रतिस्थापित करने पर,

(e^x sec² x + tan x · e^x)dx = dt

अतः ${\displaystyle I=\int 1\cdot dt}$

= t + C = e^x tan x + C