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How do you integrate the function cos^2(x)

At first, you might think that it is possible to perform this integral simply by inspection, using the 'backwards chain rule'. This method would consist of adding one to the power, to get cos3(x), then dividing by the new power and the derivative of the function, giving you -(1/3sin(x))cos3(x). However, once you have performed an integration it is always wise to check your result by differentiating to see if you get your starting function back. In this case, it is clear that differentiating -(1/3sin(x))cos3(x) does not give cos2(x), because you have to use the quotient rule to differentiate cos3(x)/sin(x).

This means that a different approach is required to perform the integration, and that is to use the trig identity cos2x=1/2+(1/2)cos(2x) to change the integrand to something which can be integrated easily. It is then simple to integrate 1/2 +(1/2)cos(2x) using the familiar method, giving the correct answer of (1/2)x+(1/4)sin(2x)+c (not forgetting the constant of integration!).

Similarly, sin2(x) can be integrated quickly using the trig identity sin2(x)=1/2-(1/2)cos(2x), so these two identities are definitely worth memorizing!

Edward R. GCSE Maths tutor, A Level Maths tutor, Uni Admissions Test ...

11 months ago

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