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Home  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus

Integrate : $\int \large\frac{1}{(\log_ex)^2+1}$

$(a)\;\tan^{-1}(\log_e x)+c \\(b)\;\cot ^{-1} (\log _e x )+c \\(c)\; \tan^{-1}\cot^{-1} (\log _e^{x})+c \\ (d)\;None$

1 Answer

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$\int \large\frac{1}{(\log_ex)^2+1}$
with respect to $(\log_ex)$
to solve this type of question
=> $\int \large\frac{1}{(\log _e x)^2} $$d(\log _e x)$
=> $\int \large\frac{1}{x} \times \large\frac{1}{1+(\log_2x)^2}$
=>$\log_e x =t$
=> $\int \large\frac{1}{1+t^2}.dt$
=> $\tan ^{-1}(t)+c$
$\tan^{-1}(\log_e x)+c$
Hence a is the correct answer.
answered Dec 31, 2013 by meena.p