Question

$\int \frac{a}{\left(1+x^{2}\right) tan^{-1}x}dx = $

• A. $ a\, log \left|tan^{-1}x\right| + C $
• B. $ \frac{a}{2} \left(tan^{-1}x \right)^{2}+ C $
• C. $ a\, log \left(1+x^{2}\right) + C $
• D. None of these

Answer 1

Answer: (A)

$I = a \int \frac{dx}{\left(1+x^{2}\right)tan^{-1}x}$

Put $tan^{-1}x = t \frac{\Rightarrow 1}{1+x^{2}} dx = dt $

$ \Rightarrow I = a \int\frac{dt}{t }= a \,log \left|t\right| +C = a\, log \left|tan^{-1}x\right| + C$