Question 29 of 33intermediate🔧 ApplyShort Answer2 marks

Integrate the functions in Exercises 1 to 22: x2x^{2} log⁡\log x

Correct Answer

∫ x² log x dx = (x³/3) log x – x³/9 + C

Exercise: EXERCISE 7.6 | Q: 6 | (Page 39)
For More Understanding

Explanation

This question requires integration by parts. The context shows similar examples where logarithmic functions are integrated by taking log as the first function and the algebraic expression as the second function. Following the ILATE rule, log x should be the first function and x² as the second function.

Solution Steps

  1. Step 1: Apply integration by parts by taking log x as the first function and x² as the second function.

  2. Step 2: ∫ x² log x dx = log x ∫ x² dx – ∫ [(d/dx(log x)) ∫ x² dx] dx

  3. Step 3: = log x · (x³/3) – ∫ [(1/x) · (x³/3)] dx

  4. Step 4: = (x³/3) log x – ∫ (x²/3) dx

  5. Step 5: = (x³/3) log x – x³/9 + C

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