Integration using partial fractions and reverse chain ruleEdexcel A-Level Maths: Flashcards

What these 12 flashcards ask

\int\frac{f'(x)}{f(x)}\,dx?
\int\frac{1}{ax+b}\,dx?
\int\frac{2}{3x+5}\,dx?
\int\frac{x}{x^2+5}\,dx?
\int(ax+b)^n\,dx for n\ne-1?
\int\frac{2}{(2x-1)^4}\,dx?
How do you test for an \frac{f'}{f} integral?
Partial fractions form for \frac{px+q}{(x-1)(2x+3)}?
Partial fractions form with a repeated factor (2x-1)^2?
Quick way to find A in \frac{A}{x-1}+\ldots?
\int\frac{3}{(2x-1)^2}\,dx?
Combine 3\ln4-\ln13+\ln7 into one logarithm?

Exam questions on Integration using partial fractions and reverse chain rule

Two rational functions are defined for $x\ge0$ by $h(x)=\frac{2}{3x+5}$ and $k(x)=\frac{x}{x^2+5}$ .
Find the exact value of $\int_0^2 3k(x)\,\mathrm{d}x$ .2 marks
Let $f(x)=\frac{4x+11}{(x-1)(2x+3)}$ for $x>1$ .
Find the exact value of $\int_2^5 f(x)\,\mathrm{d}x$ , giving your answer as a single logarithm.2 marks
A curve $C$ has gradient $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{(2x-1)^4}$ for $x>\frac12$ , and passes through the point $(1,\,1)$ .
Find the equation of $C$ .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).