(6-7)   Our mission is to find the value of a which makes this expression true:

It’s not immediately obvious how best to go about this: sin2x is not a particularly straightforward integration. If we could find an alternative way of expressing this without the squaring, it would be much easier. The key, therefore, is to go to your trigonometric identities.

①  Recall your double angle formulae and write down the expression for cos(2x). Now stretch your powers of recall still further and write down what you get if you add together cos2x + sin2x.  Finally combine these two identities to find a function that equates to sin2x but doesn’t involving squaring the trig function.

②  Use this to rewrite equation (VI-10) in terms that might be more amenable to integration.

③  You should now be able to separate out the two terms in the integration, and evaluate the definite integral for the first one, because it is easy.

④  The second term can be sorted out using integration by substitution.  Try the substitution u = 2nπ/L. Remember that you have to transform the limits of the definite integral to put them in terms of u, as well.

⑤  Evaluate the definite integral and solve the resulting equation to find the value of the constant a.