SPM 2024 Add Maths Paper 1, Question 10Integration

Solved on video in English and Bahasa Melayu · Form 5, Integration · 4:36 video

The question

(a)

Given that d/dx(1/(x² + 1)) = g(x), find ∫[3g(x) + 1] dx.

(b)(i)

(b) Diagram 8 shows the graph x = f(y). It is given that the area of the shaded region is 9/2 unit² and ∫_0^h f(y) dy = ∫_k^m f(y) dy = 3/2. Find the value of (i) ∫_0^k f(y) dy.

(b)(ii)

(b) Using the same diagram and given information as part (b)(i) above (graph x = f(y), shaded area = 9/2 unit², ∫_0^h f(y) dy = ∫_k^m f(y) dy = 3/2), find the value of (ii) ∫_h^k x dy + ∫_k^m (5/4) f(y) dy.

Part (a)

Step 1

Since d/dx(1/(x² + 1)) = g(x), integration reverses differentiation: ∫g(x) dx = 1/(x² + 1) + c.

Why this step

Integrating undoes differentiating, so we can read ∫g(x) dx straight from the given fact. Don't differentiate 1/(x² + 1) yourself; that's the common mistake.

3 more steps, each with the reason for it, in the full solution

Final answer

3/(x² + 1) + x + c

Where students lose marks

Watch out: don't differentiate 1/(x² + 1). Since g(x) is its derivative, integrating g(x) gives 1/(x² + 1) back.

Part (b)(i)

4 more steps, each with the reason for it, in the full solution

Final answer

∫_0^k f(y) dy = 0

Where students lose marks

Watch out: the middle bump lies left of the y-axis, so its integral is −3/2, not +3/2. Using +3/2 gives 3 instead of 0.

Part (b)(ii)

4 more steps, each with the reason for it, in the full solution

Final answer

3/8

Where students lose marks

Watch out: ∫_h^k f(y) dy is −3/2, not +3/2. And 5/4 is a constant multiplier, so just multiply it by 3/2.

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