← Integration methods: past papers

AS 91579 · 2023 paper · Simpson’s rule, a trig area, a factorised differential equation

Question 3(b) Achieved

Practises Standard integrals

Opens with Simpson’s Rule on five measured offsets from a garden path, then a root quotient, a product-of-sines area, an exponential-quotient acceleration integrated to a velocity, and a differential equation that has to be factorised before it will separate.

Find int ((√x − 3)/√x) dx.

The 2023 paper, page 10 ↗ Marking schedule, page 5 ↗

Work it out

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

  1. 1 Split the fraction: divide each term on top by √x.

    (√x − 3)/√x = 1 − 3/√x = 1 − 3x−1/2

  2. 2 Integrate term by term.

    int (1 − 3x−1/2)dx = x − 3x1/2/(1/2) + c = x − 6√x + c
Answer x − 6√x + c

What the marker wanted

  • Achieved The correct integral. The schedule says the + c is not required for it.

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

✕

Common mistake

Slipping while writing 3/√x as a power. NZQA’s 2023 report lists index errors with √x among the Not Achieved answers: it is 3x−1/2.

How did it go?

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