2+ 2x/1+ 2x e^x - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

..... Miscellaneous Exercise .....
Integrate the functions in Exercises 229 to 252.

Question : 249 of 271

Marks: +1, -0

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

Solution:

Let

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

I = \int \frac{e^x(2+2\sin x \cos x)}{2\cos^2 x} \, dx

= \int e^x [\sec^2 x + \tan x] \, dx

= \int e^x (\tan x + \sec^2 x) \, dx

= \int e^x \left[ \tan x + \frac{d}{dx}\tan x \right] \, dx

= e^x \tan x + C

\because \int e^x (f(x) + f'(x)) \, dx = e^x f(x) + C

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