2023TrialMultiple choice

Q7

1 mark

Question

Which expression is equal to

\int 3\sqrt{x}\ln x\,dx?

(A) $\displaystyle 2x\sqrt{x}\left(\ln x-\frac{2}{3}\right)+c$

(B) $\displaystyle 2x\sqrt{x}\left(\ln x+\frac{2}{3}\right)+c$

(D) $\displaystyle \frac{1}{\sqrt{x}}\left(\frac{3}{2}\ln x+1\right)+c$

Try the question before revealing the answer.

Solution

Answer: A

\int 3\sqrt{x}\ln x\,dx

u=\ln x\qquad v=\frac{2}{3}x^{3/2}

u'=\frac1x\qquad v'=\sqrt{x}

=3\left[\frac23x^{3/2}\ln x-\int\frac23x^{3/2}\times\frac1x\,dx\right]

=2x^{3/2}\ln x-2\int\sqrt{x}\,dx

=2x^{3/2}\ln x-2\times\frac23x^{3/2}+C

=2x^{3/2}\left(\ln x-\frac23\right)+C

$(A)$

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Syllabus alignment

Calculus

Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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