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Evaluate \( \int_{\frac{\pi}{2}}^{2\pi} (5\sin x - 3\cos x) \: dx \) - Scottish Highers Maths - Question 11 - 2023
Question 11
Evaluate \( \int_{\frac{\pi}{2}}^{2\pi} (5\sin x - 3\cos x) \: dx \). The diagram in your answer booklet shows the graphs with equations \( y = 5\sin x \) and \( y ... show full transcript
Worked Solution & Example Answer:Evaluate \( \int_{\frac{\pi}{2}}^{2\pi} (5\sin x - 3\cos x) \: dx \) - Scottish Highers Maths - Question 11 - 2023
Step 1
Evaluate \( \int_{\frac{\pi}{2}}^{2\pi} (5\sin x - 3\cos x) \: dx \)
Answer
To evaluate the integral, we first integrate the expression:
[ \int (5\sin x - 3\cos x) : dx = -5\cos x - 3\sin x + C ]
Next, we will substitute the limits from ( \frac{\pi}{2} ) to ( 2\pi ):
[ \left[-5\cos(2\pi) - 3\sin(2\pi) \right] - \left[-5\cos\left(\frac{\pi}{2}\right) - 3\sin\left(\frac{\pi}{2}\right)\right] ]
Calculating these values:
[ \left[-5(1) - 0 \right] - \left[0 - 3(1)\right] = -5 + 3 = -2 ]
Thus, the value of the integral is (-2).
Step 2
On the diagram in your answer booklet, shade the area represented by the integral in (a).
Answer
To correctly shade the area represented by the integral ( \int_{\frac{\pi}{2}}^{2\pi} (5\sin x - 3\cos x) : dx ), identify the bounds defined by ( x = \frac{\pi}{2} ) and ( x = 2\pi ). The area to shade is the region between the curves ( y = 5\sin x ) and ( y = 3\cos x ) over this interval.