Integration using partial fractionsEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $f(x)=\frac{5}{(x+1)(x^2+4)}$, which is to be written in the form $\frac{A}{x+1}+\frac{Bx+C}{x^2+4}$.

   (a)

   Why is the numerator over $x^2+4$ written as $Bx+C$ rather than as a single constant?

   [1 mark]

   • ABecause $x^2+4$ factorises into two linear factors
   • BBecause the denominator contains a repeated factor
   • CBecause the fraction is improper
   • DBecause $x^2+4$ has no real linear factors, so its numerator may be linear

   (b)

   Find the value of $A$.

   [1 mark]

   • A$1$
   • B$5$
   • C$\frac15$
   • D$-1$

   (c)

   Given that $A=1$, find the values of $B$ and $C$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $g(x)=\frac{1-x}{x^2+4}$.

   (a)

   Find $\int\frac{x}{x^2+4}\,dx$.

   [1 mark]

   • A$\ln(x^2+4)+c$
   • B$\frac12\arctan\frac{x}{2}+c$
   • C$\frac12\ln(x^2+4)+c$
   • D$\frac{x^2}{2(x^2+4)}+c$

   (b)

   Find $\int\frac{1}{x^2+4}\,dx$.

   [1 mark]

   • A$\arctan\frac{x}{2}+c$
   • B$\frac12\arctan\frac{x}{2}+c$
   • C$2\arctan\frac{x}{2}+c$
   • D$\frac12\ln(x^2+4)+c$

   (c)

   Hence find the exact value of $\int_0^2 g(x)\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\frac{2x^2+17}{(x+2)(4x^2+9)}$.

   (a)

   Express $f(x)$ in the form $\frac{A}{x+2}+\frac{Bx+C}{4x^2+9}$, where $A$, $B$ and $C$ are constants to be found.

   [3 marks]

   (b)

   Hence find the exact value of $\int_0^{3/2}f(x)\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=\frac{x+21}{(x+1)(x^2+9)}$ and $g(x)=\frac{3x^2+x+3}{(x+1)(4x^2+1)}$ for $x\ge0$.

   (a)

   Express $f(x)$ in partial fractions and hence find the exact area of the region bounded by the curve $y=f(x)$, the $x$-axis and the lines $x=0$ and $x=3$.

   [6 marks]

   (b)

   Show that $\int_0^{1/2}g(x)\,dx=\ln\frac32+\frac\pi4-\frac18\ln2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions