JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(\int \limits_{\frac{1}{3}}^3\left|\log _e x\right| d x=\frac{m}{n} \log _e\left(\frac{n^2}{e}\right)\), where \(m\) and \(n\) are coprime natural numbers, then \(m ^2+ n ^2-5\) is equal to \(............\).
- A \(20\)
- B \(21\)
- C \(22\)
- D \(24\)
Answer & Solution
Correct Answer
(A) \(20\)
Step-by-step Solution
Detailed explanation
\(\int \limits_{\frac{1}{3}}^3|\operatorname{nx}| dx =\int \limits_{\frac{1}{3}}^1(-\ell nx ) dx +\int_1^3(\ell nx ) dx\) \(=-[ x \ell nx - x ]_{\ell / 3}^1+[ x \ell nx - x ]_1^3\) \(=-\left[-1-\left(\frac{1}{3} \ell \ln \frac{1}{3}-\frac{1}{3}\right)\right]+[3 \ln 3-3-(-1)]\)…
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