IB Maths AA HL Ahl 5 16 Hl Advanced Integration Techniques Questions

Practise IB Mathematics HL 5.16 by applying advanced integration techniques methods to questions from the Question Bank.

Exam points

Use integration by substitution when the provided substitution converts the integrand into a simpler standard form.
Apply integration by parts, including repeated integration by parts, to products of functions and verify constants and bounds from complete mark-scheme evidence.

[Maximum number: 4]

The following question uses Maclaurin series to investigate approximations of mathematical constants and the accuracies of such approximations.

By using integration by parts, show that $\int_{0}^{\frac{1}{\sqrt{3}}} \arctan x \mathrm{~d} x=\frac{\pi}{6 \sqrt{3}}-\frac{1}{2} \ln \frac{4}{3}$ .

$\quad u=\arctan x$ and $\frac{\mathrm{d} v}{\mathrm{~d} x}=1$ (or equivalent)

\int_{0}^{\frac{1}{\sqrt{3}}} \arctan x \mathrm{~d} x=[x \arctan x]_{0}^{\frac{1}{\sqrt{3}}}-\int_{0}^{\frac{1}{\sqrt{3}}} \frac{x}{1+x^{2}} \mathrm{~d} x

Note: Award A1 for each term.

\begin{aligned} & =[x \arctan x]_{0}^{\frac{1}{\sqrt{3}}}-\frac{1}{2}\left[\ln \left(1+x^{2}\right)\right]_{0}^{\frac{1}{\sqrt{3}}}\left(=\frac{1}{\sqrt{3}} \arctan \frac{1}{\sqrt{3}}-\frac{1}{2} \ln \frac{4}{3}\right) \\ & \quad=\frac{\pi}{6 \sqrt{3}}-\frac{1}{2} \ln \frac{4}{3} \end{aligned}

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