CAIE A-Level Mathematics 3.5.3 Integration by Partial Fractions

Practise decomposing rational functions and integrating linear factors to logarithms or irreducible quadratic factors to inverse tangents before applying exact limits.

Exam points

decompose the rational function fully before integrating each partial-fraction term
integrate linear denominators as logarithms and quadratic forms as inverse tangents
apply transformed or original bounds and combine logarithms into the required exact form

[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

M1

The terms used need to have been obtained correctly.
Must be exact values, not decimals.

Obtain $1-\ln 3$

A1

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3.5.3—Partial fractions question 1