Pearson Edexcel IAL Mathematics P1.5.2 Integration of powers of x & related sums

Practise integrating sums of powers of x, including roots, reciprocal powers and expanded products, with answers simplified.

Exam points

rewrite roots and reciprocals as powers before integrating term by term
use a point on the curve to determine c after finding f(x) from f'(x)

[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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P1.5.2 - Integration of powers of x and related sums question 1