A student claims that

MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration

Problem

\displaystyle\int (2x)(x^{2}+1)^{5}\,dx=(x^{2}+1)^{6}+C

. After differentiating

(x^{2}+1)^{6}+C

, what conclusion should the student reach? (a) The claim is incorrect; the derivative is

6(x^{2}+1)^{5}

, which is missing the factor

2x

. (b) The claim is correct, because any multiplicative constant can be absorbed into

C

. (c) The claim is incorrect; the derivative is

12x(x^{2}+1)^{5}

, which contains an extra factor of

6

. (d) The claim is incorrect; the derivative is

12x(x^{2}+1)^{6}

, so the exponent of

(x^{2}+1)

is one too high.