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Home  >>  CBSE XII  >>  Math  >>  Integrals
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$\Large \int \normalsize\frac{\large e^x(1+x)}{\large \cos^2(e^xx)}dx$ equals

$\begin{array}{1 1} (A)\;-\cot(ex^x)+C \\(B)\;\tan(xe^x)+C \\ (C)\;\tan(e^x)+C\\ (D)\;\cot(e^x)+C\end{array}$

Can you answer this question?
 
 

1 Answer

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Toolbox:
  • $(i)\;\int e^{-x}=e^{-x}+c.$
  • $(ii)\;\int\sec^2xdx=\tan x+c.$
Given $I=\int\frac{e^x(1+x)}{\cos^2(e^xx)}dx.$
 
Let $e^xx=t.$
 
On differentiating we get,
 
$(e^x.x+e^x.1)dx=dt.$
 
$\Rightarrow e^x(x+1)dx=dt.$
 
On substituting we get,
 
Hence $I=\int\frac{dt}{\cos^2t}$.
 
$\frac{1}{\cos^2t}=\sec^2t.$
 
Therefore $I=\int\sec^2t.dt.$
 
On integrating we get,
 
$\;\;\;=\tan t+c.$
 
substituting for t we get,
 
$\;\;\;=\tan(e^xx)+c.$
 
Hence the correct answer is B.

 

answered Feb 3, 2013 by sreemathi.v
 
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