Integral Calculus Question 458
Question: A primitive of $ \frac{x}{x^{2}+1} $ is
[SCRA 1996]
Options:
A) $ {\log_{e}}(x^{2}+1) $
B) $ x{{\tan }^{-1}}x $
C) $ \frac{{\log_{e}}(x^{2}+1)}{2} $
D) $ \frac{1}{2}x{{\tan }^{-1}}x $
Show Answer
Answer:
Correct Answer: C
Solution:
$ f(x)=\frac{x}{1+x^{2}} $ ,
$ \therefore ,I=\int_{{}}^{{}}{f(x)}=\int_{{}}^{{}}{\frac{x}{1+x^{2}},dx} $ Put $ 1+x^{2}=t\Rightarrow 2x,dx=dt\Rightarrow x,dx=dt/2 $
$ \therefore ,I=\frac{1}{2}\int_{{}}^{{}}{\frac{dt}{t}=\frac{1}{2}\log t+c} $ ; $ I=\frac{1}{2}\log (1+x^{2})+c $ .
BETA
