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STEP Support Programme

1988 S2 Q7

For the part where you have to show that the integral of the function of theta from 0 to pi/4 is equal to two times the integral from 0 to pi/8. Is it enough to show that the function of theta is symmetrical about theta=pi/8 and then deduce the result or do you have to use a substitution and show it through manipulation of the integral?

For the deduction at the end of the question you have to use the substitution x=u^2 to get the left hand side, my question is how is it justified to then set u=x?

To show $$\int^{\pi/4}_0 \frac{\mathrm{d}\theta}{\left[\sqrt{2}\cos \theta \cos (\pi/4-\theta)\right]^{k+1}}=2\int^{\pi/8}_0 \frac{\mathrm{d}\theta}{\left[\sqrt{2}\cos \theta \cos (\pi/4-\theta)\right]^{k+1}},$$ I think it would be sufficient to say the integrand is symmetric about $\theta=\pi/8$ so long as you write it clearly.

Remember that the variable which you are integrating with respect to is a "dummy variable" so you can change it to any symbol you like, for example: $$\int^a_b f(x)\,\mathrm{d}x=\int^a_b f(£)\, \mathrm{d}£.$$

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