Partial fractions and reduction formulaeAQA A-Level Further Maths: Subtopic test

1. 1
   Let $\mathrm{f}(x)=\dfrac{5x+1}{(x-1)(x+2)}$.

   (a)

   Which expression is equal to $\mathrm{f}(x)$?

   [1 mark]

   • A$\dfrac{2}{x-1}+\dfrac{3}{x+2}$
   • B$\dfrac{3}{x-1}+\dfrac{2}{x+2}$
   • C$\dfrac{2}{x-1}-\dfrac{3}{x+2}$
   • D$\dfrac{5}{x-1}+\dfrac{1}{x+2}$

   (b)

   Find $\displaystyle\int\mathrm{f}(x)\,dx$.

   [1 mark]

   • A$\ln|(x-1)(x+2)|+c$
   • B$-\dfrac{2}{(x-1)^2}-\dfrac{3}{(x+2)^2}+c$
   • C$2\ln|x-1|+3\ln|x+2|+c$
   • D$3\ln|x-1|+2\ln|x+2|+c$

   (c)

   Find the exact value of $\int_2^3\mathrm{f}(x)\,dx$, giving your answer as a single logarithm.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\mathrm{g}(x)=\dfrac{5x^2+4x+9}{(x+1)(x^2+4)}$, and let $\mathrm{g}(x)\equiv\dfrac{A}{x+1}+\dfrac{Bx+C}{x^2+4}$.

   (a)

   Find the value of $A$.

   [1 mark]

   • A$10$
   • B$2$
   • C$\dfrac12$
   • D$5$

   (b)

   Given that $B=3$ and $C=1$, find $\displaystyle\int\frac{Bx+C}{x^2+4}\,dx$.

   [1 mark]

   • A$3\ln(x^2+4)+\arctan\dfrac{x}{2}+c$
   • B$\dfrac32\ln(x^2+4)+\arctan\dfrac{x}{2}+c$
   • C$\dfrac32\ln(x^2+4)+\dfrac12\arctan x+c$
   • D$\dfrac32\ln(x^2+4)+\dfrac12\arctan\dfrac{x}{2}+c$

   (c)

   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]

   Total for question 2: 4 marks

3. 3
   For integer $n\ge0$, let $I_n=\int_0^1x^n\mathrm{e}^x\,dx$.

   (a)

   Show that $I_n=\mathrm{e}-nI_{n-1}$ for $n\ge1$.

   [3 marks]

   (b)

   Show that $I_0=\mathrm{e}-1$ and hence find the exact value of $I_3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For integer $n\ge0$, let $I_n=\int_0^{\pi/2}\sin^nx\,dx$.

   (a)

   Show that, for $n\ge2$, $I_n=\dfrac{n-1}{n}I_{n-2}$.

   [6 marks]

   (b)

   Hence find the exact value of $\int_0^{\pi/2}\sin^5x\,(1+\sin x)\,dx$.

   [6 marks]

   Total for question 4: 12 marks

End of questions