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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate : $ \int \sec^4 x dx$

$(a)\;\tan x +c \qquad(b)\;\tan x +\frac{\tan ^3 x }{3} +c\qquad(c)\;\sec x+c\qquad (d)\;\cot x +c $
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1 Answer

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$\sec^2 x (1+\tan ^2 x) dx$
$\qquad=\sec^2 x dx+ \int \sec^2 x + \tan ^2 x dx$
$\qquad= \tan x + \int \sec^2 x \tan ^2 x dx$
$\tan x =t$
differentiate $\sec^2 x dx=dt$
$\qquad= \tan x +\int t^2 dt$
$\qquad= \tan x +\frac{ t^3 }{3}+c$
=>$\tan x +\large\frac{\tan ^3 x }{3} +c$
Hence b is the correct answer.
answered Dec 23, 2013 by meena.p
 
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