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Choose the correct answer in $\int e^x\sec x(1+\tan x)\;dx$ equals

\begin{array}{1 1}(A)\;e^x\cos x+C\qquad(B)\;e^x\sec x+C\\(C)\;e^x\sin x+C\qquad(D)\;e^x\tan x+C\end{array}

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1 Answer

Toolbox:
  • (i) $\int e^x[f(x)+f'(x)]dx=e^x(f(x))+c$
  • (ii) $ \frac{d}{dx}(\sec x)=\sec x \tan x$
Given $I=\int e^x \sec x(1+\tan x)dx$
 
$\qquad=\int e^x [\sec x+\sec x tan x]dx$
 
Let $f(x)=\sec x$
 
On differentiating with respect to x,
 
$f'(x)=\sec x \tan x$
 
Therefore The integral function of the form
 
$\int e^x[f(x)+f'(x)]=e^x(f(x))+c$
 
Therefore $\int e^x[\sec x+\sec x \tan x ]dx=e^x \sec x +c$
 
Hence the correct answer is B

 

answered Feb 7, 2013 by meena.p
 
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