Problem

Answer the questions given below, each carrying separate marks: a. (i) Define an antiderivative of a real-valued function. What is meant by the constant of integration? b. (ii) Give a reason why the rule

\displaystyle \int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C

cannot be applied when

n=-1

. c. (iii) Starting from the derivative

\dfrac{d}{dx}\bigl(x^{n+1}\bigr)=(n+1)x^{n}

for

n\neq-1

, derive the general formula for the antiderivative of

x^{n}

. State clearly each step used in your derivation.

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