Consider the following integral involving a general power of a linear expression.

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Consider the following integral involving a general power of a linear expression. (i) Derive a simplified expression for

\displaystyle \int (ax+b)^{n}\,dx

, where

a\ne 0

and

n\ne -1

, writing your answer in terms of

a

b

, and

n

. Do not omit the constant of integration. (ii) Using the result obtained in part (i), evaluate

\displaystyle \int (2x+3)^{2}\,dx

. (iii) Differentiate your answer from part (ii) to verify that it is indeed an antiderivative of

(2x+3)^{2}