2010 Paper 2 Q10 - partial fractions and integrals

Given that

where A, B and C are constants, find the value of A and B and show that C = 0.

(ii) Differentiate ln(x² + 4) with respect to x.

(iii) Using the results from parts (i) and (ii), find

(i)  Since

Equating coefficients of x², we have

A + 2B = 3   --- (1)

Equating coefficients of x, we get

B + 2C = 4   --- (2)

Equating constants, we have

4A + C = -20   ----(3)

From (1), A = 3 – 2B --- (4)

Sub (4) into (3), we get     4(3 – 2B) + C = - 20

                                                12 – 8B + C = -20

                                                -8B + C = -32 ---(5)

                                                B + 2C = 4  --- (2)

(5) x 2                                    -16B + 2C = -64 --- (6)

(2) – (6)                                                17B = 68

                                                B = 4

Sub B = 4 into (2), we get C = 0 (shown)

From (3)   4A = -20

                A = -5

Hence, A = -5, B = 4 and C = 0

(ii)

(iii) So,

From (ii),

so finally,