Integration using partial fractionsAQA A-Level Maths: Subtopic test

1. 1
   Let $\frac{5x+1}{(x-1)(x+2)}\equiv\frac{A}{x-1}+\frac{B}{x+2}$, where $A$ and $B$ are constants.

   (a)

   Find the values of $A$ and $B$.

   [1 mark]

   • A$A=3,\ B=2$
   • B$A=2,\ B=3$
   • C$A=2,\ B=-3$
   • D$A=6,\ B=-9$

   (b)

   Find $\int\frac{5x+1}{(x-1)(x+2)}\,dx$.

   [1 mark]

   • A$-\frac{2}{(x-1)^2}-\frac{3}{(x+2)^2}+c$
   • B$2\ln|x-1|-3\ln|x+2|+c$
   • C$2\ln|x-1|+3\ln|x+2|+c$
   • D$\frac52\ln|x^2+x-2|+c$

   (c)

   Hence find the exact value of $\int_2^4\frac{5x+1}{(x-1)(x+2)}\,dx$, giving your answer in the form $a\ln3+b\ln2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   It is given that $\frac{4}{(2x-1)(2x+1)}\equiv\frac{2}{2x-1}-\frac{2}{2x+1}$, and $x>\frac12$.

   (a)

   Find $\int\frac{2}{2x-1}\,dx$.

   [1 mark]

   • A$2\ln|2x-1|+c$
   • B$4\ln|2x-1|+c$
   • C$\frac12\ln|2x-1|+c$
   • D$\ln|2x-1|+c$

   (b)

   Find $\int\frac{4}{(2x-1)(2x+1)}\,dx$.

   [1 mark]

   • A$\ln\left|\frac{2x-1}{2x+1}\right|+c$
   • B$\ln|(2x-1)(2x+1)|+c$
   • C$2\ln|2x-1|-2\ln|2x+1|+c$
   • D$\ln\left|\frac{2x+1}{2x-1}\right|+c$

   (c)

   Hence find the exact value of $\int_1^2\frac{4}{(2x-1)(2x+1)}\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\frac{11x-1}{(x-2)(3x+1)}$ for $x>2$.

   (a)

   Express $f(x)$ in the form $\frac{A}{x-2}+\frac{B}{3x+1}$, where $A$ and $B$ are constants to be found.

   [3 marks]

   (b)

   Find the exact value of $\int_3^7f(x)\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=\frac{12}{(x+1)(x+3)}$ for $x\ge0$. The region $R$ is bounded by $C$, the $x$-axis, the $y$-axis and the line $x=3$.

   (a)

   Show that the area of $R$ is $6\ln2$.

   [6 marks]

   (b)

   The line $x=k$, where $0<k<3$, divides $R$ into two parts of equal area. Find the exact value of $k$.

   [6 marks]

   Total for question 4: 12 marks

End of questions