Find - e x -1-sin 2 x -1-cos 2 x dx

Let I = ∫ e^x √(1+sin 2x)/1 + cos 2xdx = ∫ e^x √((sin x + cos x)^2)/2 cos^2 xdx = 1/2∫ e^x (sin x + cos x/cos^2 x)dx ⇒ I = 1/2∫ e^x (sec x + sec x tan x) dx = 1 ...

Byju's Answer

Standard XII

Mathematics

Definition of Functions

Find ∫ e x √1...

Question

Find $\int e^{x} \frac{\sqrt{1 + s i n 2 x}}{1 + c o s 2 x} d x .$

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Solution

Let $I = \int e^{x} \frac{\sqrt{1 + s i n 2 x}}{1 + c o s 2 x} d x = \int e^{x} \frac{\sqrt{(s i n x + c o s x)^{2}}}{2 c o s^{2} x} d x = \frac{1}{2} \int e^{x} (\frac{s i n x + c o s x}{c o s^{2} x}) d x$
$\Rightarrow I = \frac{1}{2} \int e^{x} (s e c x + s e c x t a n x) d x = \frac{1}{2} e^{x} s e c x + C .$

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