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  4. ∫ (1+tan x /ex-cos x) dx is equal to

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Q. $\int \frac {1+tan\,x} {e^x-cos\,x} dx $ is equal to

KCETKCET 2005Integrals

Solution:

Let $I=\int \frac{1+\tan x}{e^{-x} \cos x}\, d x $
$=\int\, e^{x} \sec x \,d x+\int e^{x} \sec\, x \tan \,x \,d x$
$=e^{x} \sec x-\int e^{x} \sec\, x \tan\, x\,d x $
$+\int e^{x} \,\sec \,x \tan \,x d x+c $
$=e^{x}\,\sec\, x+c $