Question 7 - A-Level Maths

Exam Board	CAIE
Module	P2 (Pure Mathematics 2)
Year	2006
Session	June
Topic	Standard Integrals and Reverse Chain Rule

Differentiate $\ln ( 2 x + 3 )$.
Hence, or otherwise, show that $$\int _ { - 1 } ^ { 3 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x = \ln 3$$
Find the quotient and remainder when $4 x ^ { 2 } + 8 x$ is divided by $2 x + 3$.
Hence show that $$\int _ { - 1 } ^ { 3 } \frac { 4 x ^ { 2 } + 8 x } { 2 x + 3 } d x = 12 - 3 \ln 3$$