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 ...

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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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\int \frac{sin 2 x}{1 + {cos}^{2} x} d x

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I = \int e^{x} \frac{(2 + sin 2 x)}{(1 + cos 2 x)} d x .

\int (sin 2 x - cos 2 x) d x = \frac{1}{\sqrt{(k)}} sin (2 x - a) + b .

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Find ∫ex√1+sin 2x1+cos 2xdx.

Let I=∫ex√1+sin 2x1+cos 2xdx=∫ex√(sin x+cos x)22 cos2xdx=12∫ex(sin x+cos xcos2x)dx ⇒I=12∫ex(sec x+sec x tan x)dx=12exsec x+C.

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