June 2025 Paper 1 Q5

AQACurrent spec3 marksIntegration

5 Find \(\displaystyle\int \left(4x - 3 + x^{-\frac{1}{2}}\right)\mathrm{d}x\) [3 marks]

Mark scheme

Scheme	Marks	AO
Integrates at least one term correctly, may be unsimplified	M1	1.1a
Integrates at least two terms correctly, may be unsimplified	M1	1.1a
Obtains \(2x^2 - 3x + 2x^{\frac{1}{2}} + c\) or \(2x^2 - 3x + 2\sqrt{x} + c\) or \(2x^2 - 3x + 2x^{0.5} + c\)	A1	1.1b
	(3 marks)

Typical solution

\[\int \left(4x - 3 + x^{-\frac{1}{2}}\right)\mathrm{d}x\]\[= \frac{4x^2}{2} - 3x + \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + c\]\[= 2x^2 - 3x + 2x^{\frac{1}{2}} + c\]