Partial fractions and reduction formulaeAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

Partial fraction form for \dfrac{px+q}{(x-a)(x-b)}?
Partial fraction form when the denominator contains x^2+c?
\displaystyle\int\frac{k}{ax+b}\,dx?
\displaystyle\int\frac{x}{x^2+c}\,dx?
\displaystyle\int\frac{1}{x^2+c}\,dx for c0?
Two ways to find partial fraction constants?
What is a reduction formula?
Reduction formula for \int0^1x^ne^x\,dx?
Reduction formula for \int0^{\pi/2}\sin^nx\,dx?
Technique used to derive most reduction formulae?
Values of I0 and I1 for \int0^{\pi/2}\sin^nx\,dx?
How are \ln p+\ln q and k\ln p combined?

Exam questions on Partial fractions and reduction formulae

Let $\mathrm{f}(x)=\dfrac{5x+1}{(x-1)(x+2)}$ .
Find the exact value of $\int_2^3\mathrm{f}(x)\,dx$ , giving your answer as a single logarithm.2 marks
Let $\mathrm{g}(x)=\dfrac{5x^2+4x+9}{(x+1)(x^2+4)}$ , and let $\mathrm{g}(x)\equiv\dfrac{A}{x+1}+\dfrac{Bx+C}{x^2+4}$ .
Find the exact value of $\displaystyle\int_0^2\frac{3x}{x^2+4}\,dx$ , giving your answer in the form $p\ln q$ .2 marks
For integer $n\ge0$ , let $I_n=\int_0^1x^n\mathrm{e}^x\,dx$ .
Show that $I_n=\mathrm{e}-nI_{n-1}$ for $n\ge1$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).