Selangor Trial 2025 Add Maths Paper 1, Question 9Integration

Solved on video in English · Form 5, Integration · 4:58 video

The question

(a)

(a) It is given that y = (2x + 5)³/(6x − 1) and dy/dx = 12(2x + 5)²(2x − 3)/(6x − 1)². Find ∫ [3(2x + 5)²(2x − 3)/(6x − 1)²] dx in terms of x.

(b)

(b) Diagram 6 shows a curve y = ½x² + 1 and its tangent y = 2x − 1 at the point A(2, 3). Find the area of the shaded region.

Step 1

Notice the integrand is a constant multiple of dy/dx: 3(2x + 5)²(2x − 3)/(6x − 1)² = (1/4) × 12(2x + 5)²(2x − 3)/(6x − 1)² = (1/4)(dy/dx).

Why this step

The messy factor matches the given dy/dx, so don't integrate the quotient directly. Only the front number differs: 3 is a quarter of 12.

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

Final answer

(2x + 5)³/[4(6x − 1)] + c

Where students lose marks

Watch out: the integrand is (1/4)(dy/dx), so the integral is y/4, not 4y. Do not forget + c.

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

Final answer

13/12 (≈ 1.083) unit²

Where students lose marks

Watch out: the tangent crosses the x-axis at x = ½, ∫ ½x² dx = x³/6, and 10/3 − 9/4 needs denominator 12.

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