Question 9 of 33intermediate🔧 ApplyShort Answer2 marks

Integrate the functions in Exercises 1 to 37: sin⁡\sin (ax + b) cos⁡\cos (ax + b)

Correct Answer

sin⁡2(ax+b)2a\frac{\sin^2(ax+b)}{2a} + C

Exercise: EXERCISE 7.2 | Q: 5 | (Page 16)
For More Understanding

Explanation

The solution applies the substitution method demonstrated in the provided context. First, the linear argument is substituted (similar to 'Put x + a = t' in Chunk 2). Next, the trigonometric term is substituted (similar to 'Put cos x = t' in Chunk 2) to reduce the integral to a standard power form.

Solution Steps

  1. Step 1: Let ax+b=tax + b = t. Then adx=dta dx = dt or dx=1adtdx = \frac{1}{a} dt.

  2. Step 2: Substitute into the integral: I=1a∫sin⁡tcos⁡tdtI = \frac{1}{a} \int \sin t \cos t dt.

  3. Step 3: Let sin⁡t=u\sin t = u. Then cos⁡tdt=du\cos t dt = du.

  4. Step 4: The integral becomes 1a∫udu=1a[u22]+C=u22a+C\frac{1}{a} \int u du = \frac{1}{a} \left[ \frac{u^2}{2} \right] + C = \frac{u^2}{2a} + C.

  5. Step 5: Substitute back for uu and tt: I=sin⁡2(ax+b)2a+CI = \frac{\sin^2(ax+b)}{2a} + C.

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