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

1. 1
   Two rational functions are defined for $x\ge0$ by $h(x)=\frac{2}{3x+5}$ and $k(x)=\frac{x}{x^2+5}$.

   (a)

   Find $\int h(x)\,\mathrm{d}x$.

   [1 mark]

   • A$2\ln|3x+5|+c$
   • B$\frac23\ln|3x+5|+c$
   • C$6\ln|3x+5|+c$
   • D$-\frac{2}{(3x+5)^2}+c$

   (b)

   Find $\int k(x)\,\mathrm{d}x$.

   [1 mark]

   • A$\ln(x^2+5)+c$
   • B$2\ln(x^2+5)+c$
   • C$\frac12\ln(x^2+5)+c$
   • D$\frac{x^2}{2}\ln(x^2+5)+c$

   (c)

   Find the exact value of $\int_0^2 3k(x)\,\mathrm{d}x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=\frac{4x+11}{(x-1)(2x+3)}$ for $x>1$.

   (a)

   Which expression is equal to $f(x)$?

   [1 mark]

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

   (b)

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

   [1 mark]

   • A$3\ln(x-1)-2\ln(2x+3)+c$
   • B$3\ln(x-1)+2\ln(2x+3)+c$
   • C$-\frac{3}{(x-1)^2}+\frac{4}{(2x+3)^2}+c$
   • D$3\ln(x-1)-\ln(2x+3)+c$

   (c)

   Find the exact value of $\int_2^5 f(x)\,\mathrm{d}x$, giving your answer as a single logarithm.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   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 $(1,\,1)$.

   (a)

   Find the equation of $C$.

   [3 marks]

   (b)

   The region $R$ is bounded by $C$, the $x$-axis and the lines $x=1$ and $x=2$. Find the exact area of $R$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=\frac{6x^2-6x+6}{(x+1)(2x-1)^2}$ for $x>\frac12$.

   (a)

   Express $f(x)$ in the form $\frac{A}{x+1}+\frac{B}{2x-1}+\frac{C}{(2x-1)^2}$, finding the constants $A$, $B$ and $C$.

   [6 marks]

   (b)

   Hence show that $\int_1^3f(x)\,\mathrm{d}x=\frac65+\ln\frac{4}{\sqrt5}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions