Consider the function g(x)=6 (x+ 6 )+ 5 (x-3)^2+4 -9 x

MATHEMATICS · Subjective · A Level · Mathematics · Integration · Integration

Problem

Consider the function g(x)=6sin⁡(x+π6)+5(x−3)2+4−9cos⁡x.g(x)=6\sin\bigl(x+\tfrac{\pi}{6}\bigr)+\frac{5}{(x-3)^2+4}-9\cos x. (i) Find a general antiderivative G(x)G(x) of g(x)g(x). (ii) Show by direct differentiation that ddxG(x)=g(x)\dfrac{d}{dx}G(x)=g(x). (iii) Using your expression for G(x)G(x), evaluate the exact value of G(π3)−G(0)G\bigl(\tfrac{\pi}{3}\bigr)-G(0).

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