1.62: Product rule and integration of a logarithm

0606/13/M/J/17 — Question 10 · 9 marks

It is given that

y = (10x + 2)\ln(5x + 1)

.

(i)

Find

\dfrac{\mathrm{d}y}{\mathrm{d}x}

.

[4]

(ii)

Hence show that

\displaystyle\int \ln(5x + 1)\,\mathrm{d}x = \frac{(ax + b)}{5}\ln(5x + 1) - x + c

, where

a

and

b

are integers and

c

is a constant of integration.

[3]

(iii)

Hence find

\displaystyle\int_{0}^{\frac{1}{5}} \ln(5x + 1)\,\mathrm{d}x

, giving your answer in the form

\dfrac{d + \ln f}{5}

, where

d

and

f

are integers.

[2]