← Integration methods: past papers

AS 91579 · 2025 paper · standard integrals, separation of variables

Question 1(b) Achieved

Practises Differential equations

Five independent parts, no shared scenario: a standard integral, a differential equation with a boundary condition, a pair of definite integrals that pin down two constants, a trig area, and a differential equation you have to rearrange before you can separate it.

Solve the differential equation dy/dx = 3√x + 2/√x, given that y = 10 when x = 4.

The 2025 paper, page 2 ↗ Marking schedule, page 1 ↗

Work it out

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

  1. 1 Before integrating, write each term as a power of x.

    3√x = 3x1/2 and 2/√x = 2x−1/2, so dy/dx = 3x1/2 + 2x−1/2.

  2. 2 Integrate term by term: add one to each power, then divide by the new power.

    y = 3x3/2/(3/2) + 2x1/2/(1/2) + c = 2x3/2 + 4x1/2 + c

  3. 3 Use y = 10 when x = 4 to find c. What are 41/2 and 43/2?

    41/2 = 2 and 43/2 = 8, so 10 = 2(8) + 4(2) + c = 24 + c, and c = −14.
Answer y = 2x3/2 + 4√x − 14

What the marker wanted

  • Achieved The whole solution, y = 2x3/2 + 4x1/2 − 14, with the integration shown.

In our words, from NZQA’s 2025 marking schedule.

✕

Common mistake

Integrating the roots without rewriting them as powers. NZQA’s 2025 report lists not seeing that √x is x1/2, or that 1/√x is x−1/2, among the Not Achieved answers.

How did it go?

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