1.2: Differentiation by product rule and definite integration

0606/11/M/J/16 — Question 5 · 8 marks

Do not use a calculator in this question.

(i)

Show that

\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\mathrm{e}^{4x}}{4} - x\mathrm{e}^{4x}\right) = px\mathrm{e}^{4x}

, where

p

is an integer to be found.

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(ii)

Hence find the exact value of

\displaystyle\int_{0}^{\ln 2} x\mathrm{e}^{4x}\,\mathrm{d}x

, giving your answer in the form

a\ln 2 + \dfrac{b}{c}

, where

a

b

and

c

are integers to be found.

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