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Find $$\int 7 \cos \left( 4x + \frac{\pi}{3} \right) dx.$$ - Scottish Highers Maths - Question 3 - 2023

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Find-$$\int-7-\cos-\left(-4x-+-\frac{\pi}{3}-\right)-dx.$$-Scottish Highers Maths-Question 3-2023.png

Find $$\int 7 \cos \left( 4x + \frac{\pi}{3} \right) dx.$$

Worked Solution & Example Answer:Find $$\int 7 \cos \left( 4x + \frac{\pi}{3} \right) dx.$$ - Scottish Highers Maths - Question 3 - 2023

Step 1

Start to integrate

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Answer

To begin the integration, we recognize that this is a standard integral of the form ∫acos⁡(bx+c) dx\int a \cos(bx + c) \, dx, where in this case, we have a=7a = 7, b=4b = 4, and c=π3c = \frac{\pi}{3}.

Step 2

Complete integration

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Answer

Using the formula for integrating the cosine function, we have: ∫acos⁡(bx+c) dx=absin⁡(bx+c)+C,\int a \cos(bx + c) \, dx = \frac{a}{b} \sin(bx + c) + C, where CC is the constant of integration.

Substituting the values in, we calculate: ∫7cos⁡(4x+π3)dx=74sin⁡(4x+π3)+C.\int 7 \cos \left( 4x + \frac{\pi}{3} \right) dx = \frac{7}{4} \sin \left( 4x + \frac{\pi}{3} \right) + C.

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