For a real constant

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

Problem

For a real constant

k\neq-1

, consider the function

g_k(x)=(2k+1)x^{k}

. (i) Obtain the most general antiderivative

G_k(x)

g_k(x)

in terms of

k

. Simplify your answer. (ii) Taking

k=\dfrac{3}{2}

, evaluate

G_k(4)-G_k(1)

. Express your result in simplest form. (iii) Explain briefly why the formula used in part (i) is not valid when

k=-1

. Then determine the antiderivative of

g_{-1}(x)

when

k=-1