Question 30 of 33intermediate🔧 ApplyShort Answer2 marks

Integrate the functions in Exercises 1 to 22: x sec⁡2\sec^2x

Correct Answer

To find ∫xsec⁡2x dx\int x \sec^2 x \, dx, we use integration by parts.

Taking xx as the first function and sec⁡2x\sec^2 x as the second function, we apply the formula ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du.

Here, u=xu = x and dv=sec⁡2x dxdv = \sec^2 x \, dx, so du=dxdu = dx and v=tan⁡xv = \tan x.

Therefore, ∫xsec⁡2x dx=xtan⁡x−∫tan⁡x dx\int x \sec^2 x \, dx = x \tan x - \int \tan x \, dx

From the context, ∫tan⁡x dx=log⁡∣sec⁡x∣+C\int \tan x \, dx = \log |\sec x| + C.

Hence, ∫xsec⁡2x dx=xtan⁡x−log⁡∣sec⁡x∣+C\int x \sec^2 x \, dx = x \tan x - \log |\sec x| + C.

Exercise: EXERCISE 7.6 | Q: 12 | (Page 39)
For More Understanding

Explanation

This question requires integration by parts. The context provides the standard result that ∫tan x dx = log|sec x| + C. Since sec²x is the derivative of tan x, we can directly integrate it. The integration by parts technique shown in Example 21 of the context demonstrates the method of taking one function as 'first' and another as 'second' function.

Solution Steps

  1. Step 1: Identify the need for integration by parts with u = x and dv = sec²x dx

  2. Step 2: Compute du = dx and v = tan x

  3. Step 3: Apply integration by parts formula: ∫u dv = uv - ∫v du

  4. Step 4: Substitute: x·tan x - ∫tan x dx

  5. Step 5: Use the standard result from context: ∫tan x dx = log|sec x| + C

  6. Step 6: Final answer: x tan x - log|sec x| + C

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