EXM.1.AHL.TZ0.1

pestleMathematicsAAHLPaper 1EXM· ahl-1-11-partial-fractionssource ↗

Let $f(x) = \frac{1}{{1 - {x^2}}}$ for $- 1 < x < 1$ . Use partial fractions to find $\int {f\left( x \right)} {\text{ }}dx$ .

Markscheme / solution

$\frac{1}{{1 - {x^2}}} = \frac{1}{{\left( {1 - x} \right)\left( {1 + x} \right)}} \equiv \frac{A}{{1 - x}} + \frac{B}{{1 + x}}$ M1M1A1

$\Rightarrow 1 \equiv A\left( {1 + x} \right) + B\left( {1 - x} \right) \Rightarrow A = B = \frac{1}{2}$ M1A1A1

$\int {\frac{{\tfrac{1}{2}}}{{1 - x}}} + \frac{{\tfrac{1}{2}}}{{1 + x}}dx = \frac{{ - 1}}{2}\ln \left( {1 - x} \right) + \frac{1}{2}\ln \left( {1 + x} \right) + c$ $\left( { = \ln k\sqrt {\frac{{1 + x}}{{1 - x}}} } \right)$ M1A1

[8 marks]

Examiners’ report

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