Integration using partial fractionsEdexcel International A Level Maths: Subtopic test

1. 1
   Let $f(x)=\frac{4x+5}{(x+1)(x+2)}$ for $x>-1$.

   (a)

   Express $f(x)$ in partial fractions.

   [1 mark]

   • A$\frac{3}{x+1}+\frac{1}{x+2}$
   • B$\frac{4}{x+1}+\frac{5}{x+2}$
   • C$\frac{1}{x+1}-\frac{3}{x+2}$
   • D$\frac{1}{x+1}+\frac{3}{x+2}$

   (b)

   Hence find $\int f(x)\,dx$.

   [1 mark]

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

   (c)

   Hence find $\int_0^2f(x)\,dx$, giving your answer in the form $\ln k$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C$ has gradient $\frac{dy}{dx}=\frac{2}{3x+5}+\frac{3}{(x-1)^2}$ for $x>1$.

   (a)

   Find $\int\frac{2}{3x+5}\,dx$.

   [1 mark]

   • A$2\ln(3x+5)+c$
   • B$\frac23\ln(3x+5)+c$
   • C$\frac32\ln(3x+5)+c$
   • D$-\frac{2}{3(3x+5)^2}+c$

   (b)

   Find $\int\frac{3}{(x-1)^2}\,dx$.

   [1 mark]

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

   (c)

   The curve $C$ passes through the point $(2,1)$. Find the equation of $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\frac{5x+7}{(x-1)(x+3)}$ for $x>1$.

   (a)

   Express $f(x)$ in partial fractions.

   [3 marks]

   (b)

   Hence find $\int_2^5f(x)\,dx$, giving your answer in the form $\ln k$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=\frac{2x+5}{(x+1)(x+2)^2}$ for $x>-1$.

   (a)

   Find the constants $A$, $B$ and $C$ such that $f(x)=\frac{A}{x+1}+\frac{B}{x+2}+\frac{C}{(x+2)^2}$.

   [6 marks]

   (b)

   Hence find the exact value of $\int_0^2f(x)\,dx$, giving your answer in the form $a\ln b+c$, where $a$ and $c$ are rational and $b$ is a rational number.

   [6 marks]

   Total for question 4: 12 marks

End of questions