If ∫_(1/3)³ |log ₑx|dx = m/nlog ₑ( n²/e ) , where m and n are coprime natural numbers, then m² + n² - 5 is equal to _____________.

Solution

Maths sir · 4 steps

The modulus flips sign at x = 1, so split the integral there and handle each half separately.

1. Remove the modulus. For 1/3 ≤ x < 1, ln x < 0, so |ln x| = -ln x. For 1 ≤ x ≤ 3, ln x ≥ 0, so |ln x| = ln x.

   ∫_(1/3)³ |ln x| dx = -∫_(1/3)¹ ln x dx + ∫₁³ ln x dx, ∫ ln x dx = x ln x - x

2. Evaluate each part. On [1, 3]: (3 ln 3 - 3) - (0 - 1) = 3 ln 3 - 2. On [1/3, 1]: -[(-1) - (-1/3 ln 3 - 1/3)] = 2/3 - 1/3 ln 3.
3. Match the form. Adding gives 8/3 ln 3 - 4/3 = 4/3(2 ln 3 - 1) = 4/3 ln(9/e) = 4/3 ln(3²/e). So m = 4 and n = 3, which are coprime.
4. So. m² + n² - 5 = 16 + 9 - 5 = 20.

The idea to keep. Whenever a modulus sits inside an integral, find where the inside changes sign and split the limits there. Then write 1 as ln e to pull the answer into a single logarithm.

Answer from the NTA final answer key. Solution by PrepWiser.