CAIE A-Level Further Mathematics 2.4.2 Reduction Formulae

Practise deriving a recurrence from a hinted derivative or integration by parts, then substituting the required index and exact base integral to evaluate a definite case.

Exam points

differentiate the hinted product or integrate by parts and rewrite the remaining power in I notation
evaluate the boundary term explicitly before rearranging to the stated recurrence relation
insert the requested index and an exact base integral to calculate the target definite integral

[Maximum number: 5]

The integral $I_{n}$ , where n is an integer, is defined by $I_{n}=\int_{0}^{\frac{4}{3}}\left(1+x^{2}\right)^{\frac{1}{2} n} \mathrm{~d} x$ .

By considering $\frac{\mathrm{d}}{\mathrm{d} x}\left(x\left(1+x^{2}\right)^{\frac{1}{2} n}\right)$ , or otherwise, show that

(n+1) I_{n}=n I_{n-2}+\frac{4}{3}\left(\frac{5}{3}\right)^{n} .

$\frac{\mathrm{d}}{\mathrm{d} x}\left(x\left(1+x^{2}\right)^{\frac{1}{2} n}\right)=n x^{2}\left(1+x^{2}\right)^{\frac{1}{2} n-1}+\left(1+x^{2}\right)^{\frac{1}{2} n}$

M1 A1

Uses the product rule to differentiate.

$n\left(1+x^{2}-1\right)\left(1+x^{2}\right)^{\frac{1}{2} n-1}+\left(1+x^{2}\right)^{\frac{1}{2} n}$

M1*

Uses $x^{2}=1+x^{2}-1$ .

$\left[x\left(1+x^{2}\right)^{\frac{1}{2} n}\right]_{0}^{\frac{4}{3}}=n I_{n}-n I_{n-2}+I_{n}$

DM1

Integrates both sides using the limits given. Requires
previous method mark.

$\frac{4}{3}\left(\frac{5}{3}\right)^{n}=(n+1) I_{n}-n I_{n-2} \Rightarrow(n+1) I_{n}=n I_{n-2}+\frac{4}{3}\left(\frac{5}{3}\right)^{n}$

A1

Substitutes limits and rearranges. AG.

5

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2.4.2—And use reduction formulae for the question 1