Pearson Edexcel IAL Mathematics P1.5 Integration Question Bank

Practise reversing differentiation to integrate powers, roots and expanded expressions, including using points to recover f(x).

Exam points

integrate powers, roots and reciprocal powers of x, keeping answers in simplest form
recover f(x) from f'(x), then use a point on the curve to find the constant
expand products before integrating when the integrand is not already a sum of powers

Covered syllabus points

P1.5.1 - Indefinite integration as the reverse P1.5.2 - Integration of powers of x and related sums

[Maximum number: 7]

The curve C has equation y=f(x) where x>0

Given that
- $\mathrm{f}^{\prime}(x)=\frac{(x+3)^{2}}{x \sqrt{x}}$
- the point P(4,20) lies on C

Find f(x), simplifying your answer.

$f'(x)=\frac{(x+3)^2}{x\sqrt{x}}=\frac{x^2+6x+9}{x\sqrt{x}}$ $=x^{1/2}+6x^{-1/2}+9x^{-3/2}$ $f(x)=\frac23x^{3/2}+12x^{1/2}-18x^{-1/2}+c$ $20=\frac23(4)^{3/2}+12(4)^{1/2}-18(4)^{-1/2}+c\Rightarrow c=-\frac13$ $f(x)=\frac23x^{3/2}+12x^{1/2}-18x^{-1/2}-\frac13$

M1 A1 dM1 A1 A1 M1 A1

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Covered syllabus points

P1.5 - Integration question 1