If _ x 0 f(x) x^ 2 exists finitely and _ x 0 (1+x+ f(x) x )^ 1 / x =e^ 3 where f ( x ) =…
Mathematics · JEE Advanced · NTA Exams — Integral Calculus
If \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}\) exists finitely and
\(\lim _{x \rightarrow 0}\left(1+x+\frac{f(x)}{x}\right)^{1 / x}=e^{3}\) where f (x) = ax2+bx+c
then \(\int f(x) \log _{e} x d x\)is equal to
\(\lim _{x \rightarrow 0}\left(1+x+\frac{f(x)}{x}\right)^{1 / x}=e^{3}\) where f (x) = ax2+bx+c
then \(\int f(x) \log _{e} x d x\)is equal to
- \(\frac{2}{3} x^{3}\left(\log _{e} x-\frac{1}{3}\right)+c\)
- \(\frac{x^{3}}{3}\left(\log _{e} x-\frac{1}{3}\right)+c\)
- \(\frac{2}{3} x^{3}\left(\log _{e} x+1\right)+c\)
- \(\frac{2}{3} x^{3}\left(\log _{e} x-1\right)+c\)
Answer
(A) 2 3 x^ 3 ( _ e x- 1 3 )+c
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