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Given that \[ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} x \cos x \, dx = a n + b \] find the exact value of \( a \) and the exact value of \( b \) - AQA - A-Level Maths Mechanics - Question 8 - 2021 - Paper 3
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Given that \[ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} x \cos x \, dx = a n + b \] find the exact value of \( a \) and the exact value of \( b \). Fully justify your a... show full transcript
Worked Solution & Example Answer:Given that \[ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} x \cos x \, dx = a n + b \] find the exact value of \( a \) and the exact value of \( b \) - AQA - A-Level Maths Mechanics - Question 8 - 2021 - Paper 3
Step 1
Use integration by parts with \( u = x \) and \( v' = \cos x \)
Answer
We apply integration by parts: [ \int u , dv = uv - \int v , du ] Using ( u = x ) and ( v' = \cos x ), we find ( du = dx ) and ( v = \sin x ). Now substituting we get: [ \int x \cos x , dx = x \sin x - \int \sin x , dx ]
Step 2
Step 3
Evaluate from \( \frac{\pi}{4} \) to \( \frac{\pi}{3} \)
Answer
Now we evaluate the integral: [ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} x \cos x , dx = \left[ x \sin x + \cos x \right]_{\frac{\pi}{4}}^{\frac{\pi}{3}} ] Calculating at the limits gives: [ \left( \frac{\pi}{3} \sin \frac{\pi}{3} + \cos \frac{\pi}{3} \right) - \left( \frac{\pi}{4} \sin \frac{\pi}{4} + \cos \frac{\pi}{4} \right) ]
Step 4
Substitute values and simplify
Answer
Substituting values, we have: ( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} ), ( \cos \frac{\pi}{3} = \frac{1}{2} ), ( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} ) [ = \left( \frac{\pi}{3} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \right) - \left( \frac{\pi}{4} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \right) ] Further simplifying gives: [ = \frac{\pi \sqrt{3}}{6} + \frac{1}{2} - \frac{\pi \sqrt{2}}{8} - \frac{\sqrt{2}}{2} ]
Step 5
Identify \( a \) and \( b \)
Answer
From the integrated expression, we compare with ( an + b ) to identify:
- ( a = \frac{\sqrt{3}}{6} ) or ( \frac{\sqrt{2}}{8} )
- ( b = \left( \frac{1}{2} - \frac{\sqrt{2}}{2} \right) ) Thus, the exact values are: ( a = \frac{\sqrt{3}}{6} ) and ( b = \frac{1 - \sqrt{2}}{2} ).