Quick questions on Calculus of trigonometric functions: derivatives, integrals and harmonic motion modelling

2short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is useful identities for integration?

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Some trigonometric integrals have no direct antiderivative until you rewrite them with an identity. The most useful in Maths Advanced are the double-angle forms

\sin^2 x = \frac{1 - \cos 2x}{2}

and

\cos^2 x = \frac{1 + \cos 2x}{2}

. These convert a squared trig function (which you cannot integrate directly) into a constant plus a cosine of a double angle (which you can). For example,

\int \cos^2 x\,dx = \int \frac{1 + \cos 2x}{2}\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C

What is sign error on

\cos

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\frac{d}{dx}(\cos x) = -\sin x

and

\int \sin x \, dx = -\cos x + C

. Forgetting the minus is the single most common arithmetic slip.

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