Question 27 of 33intermediate🔧 ApplyShort Answer2 marks

Integrate the functions in Exercises 1 to 22: x log⁡\log x

Correct Answer

To find ∫xlog⁡x dx\int x \log x \, dx, we apply integration by parts. We take log⁡x\log x as the first function and xx as the second function.

Using the formula:

∫xlog⁡x dx=log⁡x⋅∫x dx−∫[ddx(log⁡x)⋅∫x dx] dx\int x \log x \, dx = \log x \cdot \int x \, dx - \int \left[\frac{d}{dx}(\log x) \cdot \int x \, dx\right] \, dx =log⁡x⋅x22−∫1x⋅x22 dx= \log x \cdot \frac{x^2}{2} - \int \frac{1}{x} \cdot \frac{x^2}{2} \, dx =x2log⁡x2−∫x2 dx= \frac{x^2 \log x}{2} - \int \frac{x}{2} \, dx =x2log⁡x2−x24+C= \frac{x^2 \log x}{2} - \frac{x^2}{4} + C

Therefore, ∫xlog⁡x dx=x22log⁡x−x24+C\int x \log x \, dx = \frac{x^2}{2} \log x - \frac{x^2}{4} + C.

Exercise: EXERCISE 7.6 | Q: 4 | (Page 39)
For More Understanding

Explanation

The question asks to integrate x log x using integration by parts. The context demonstrates this method with log x as the first function. Following the same approach, we take log x as the first function (since its derivative 1/x simplifies the integral) and x as the second function. The integral of x is x²/2, and when multiplied by 1/x (the derivative of log x), it gives x/2, which integrates to x²/4.

Solution Steps

  1. Step 1: Take log x as first function and x as second function

  2. Step 2: Apply integration by parts formula

  3. Step 3: Calculate: log x · x²/2 - ∫(1/x · x²/2)dx

  4. Step 4: Simplify: x²log x/2 - ∫(x/2)dx

  5. Step 5: Final answer: x²log x/2 - x²/4 + C

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