← Integration methods: past papers

AS 91579 · 2023 paper · an exponential integral, a rate constant, a chocolate factory

Question 2(d) Merit

Practises How to choose u

An exponential integral, a negative-power differential equation, an upper limit that appears on both sides, a trig quotient whose top is a multiple of the derivative of its bottom, and a proportional-to-the-square model built from scratch.

Find int ((cos 2x + sin 2x)/(cos 2x − sin 2x)) dx.

The 2023 paper, page 7 ↗ Marking schedule, page 3 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

  1. 1 Differentiate the bottom. How does it compare with the top?

    (d/dx)(cos 2x − sin 2x) = −2 sin 2x − 2 cos 2x = −2(cos 2x + sin 2x): the top, times −2.

  2. 2 So the top is −1/2 of the derivative of the bottom. What does that give?

    With u = cos 2x − sin 2x, (cos 2x + sin 2x) dx = −(1/2) du:
    int −(1/2u) du = −(1/2) ln |u| + c
Answer −(1/2) ln |cos 2x − sin 2x| + c

What the marker wanted

  • Achieved The integral with its minus sign missing.
  • Merit The correct integral, −(1/2) ln |cos 2x − sin 2x| + c.

In our words, from NZQA’s 2023 marking schedule.

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Common mistake

Losing the minus sign: the bottom’s derivative is −2 times the top, not 2 times. The schedule gives (1/2) ln |cos 2x − sin 2x| Achieved only.

How did it go?

Got it moves it further out. Review brings it back tomorrow.