CAIE A-Level Mathematics A2 3.5.3 Partial Fractions Questions

Practise evaluating integrals using partial fractions and related algebraic techniques.

Exam points

decompose rational integrands using partial fractions
integrate partial-fraction components accurately
apply partial fractions to definite and differential-equation integrals

[Maximum number: 5]

$8 \quad$ Let $\mathrm{f}(x)=\frac{3-3 x^{2}}{(2 x+1)(x+2)^{2}}$ .

Hence find the exact value of $\int_{0}^{4} \mathrm{f}(x) \mathrm{d} x$ , giving your answer in the form $a+b \ln c$ , where a, b and c are integers.

Integrate and obtain one of $\frac{1}{2} \ln (2 x+1),-2 \ln (x+2), \frac{-3}{x+2}$

B1 FT

The follow through is on their A, B, C.

Obtain a second term

B1 FT

If the alternative form is used, then either need to use
integration by parts or split the fraction further.

Obtain the third term

B1 FT

Substitute limits correctly in an integral with at least two terms of the form
$\frac{1}{2} \ln (2 x+1),-2 \ln (x+2)$ and $\frac{-3}{x+2}$ and subtract in correct order
The terms used need to have been obtained correctly.
Must be exact values, not decimals.

Obtain $1-\ln 3$

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CAIE A-Level Mathematics A2 3.5.3 Partial Fractions Questions question 1