If 1/3^3 | (x)| dx = mn ( n^2e), where m and n are coprime natural numbers, then m^2 + n^2 - 5 is equal to.

If 1/3^3 | (x)| dx = mn ( n^2e), where m and n are coprime natura… — Solution | JEE Mathematics

NVAMediumJEE 2023Definite Integrals

If $\int_{1/3}^{3} |\ln(x)| \, dx = \frac{m}{n} \ln\left(\frac{n^2}{e}\right)$ , where $m$ and $n$ are coprime natural numbers, then $m^2 + n^2 - 5$ is equal to _____.

Answer

Correct answer:20

Step-by-step solution

the solution unavailable for this question

Given: the solution appears unrelated to the question. It discusses a different integral and different final expressions.

Find: The value of $m^2 + n^2 - 5$ .

Since the supplied solution content does not correspond to the given question, the working could not be extracted reliably from the working. Using the provided correct answer field, the final answer is $20$ .

Common mistakes

Treating $|\ln x|$ as just $\ln x$ on the entire interval is incorrect, because $\ln x < 0$ for $0 < x < 1$ and $\ln x > 0$ for $x > 1$ . Split the integral at $x = 1$ and remove the modulus piecewise.
Forgetting to change the sign on $\left[\frac{1}{3},1\right]$ leads to a wrong value of the integral. On this interval, use $|\ln x| = -\ln x$ , not $\ln x$ .
After evaluating the integral, matching it carelessly with $\frac{m}{n}\ln\left(\frac{n^2}{e}\right)$ can cause wrong identification of $m$ and $n$ . First rewrite the result in exactly the same logarithmic form, then compare coefficients and arguments.

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