Integration involving inverse trig

A question is this type if and only if it requires integrating an inverse trigonometric function (like tan⁻¹(x)) using integration by parts.

4 questions · Standard +0.7

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CAIE P3 2021 June Q4
5 marks Standard +0.8
4 Using integration by parts, find the exact value of \(\int _ { 0 } ^ { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).
CAIE P3 2024 June Q10
10 marks Standard +0.8
10
  1. Given that \(2 x = \tan y\), show that \(\frac { d y } { d x } = \frac { 2 } { 1 + 4 x ^ { 2 } }\).
  2. Hence find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { \frac { \sqrt { 3 } } { 2 } } x \tan ^ { - 1 } ( 2 x ) \mathrm { d } x\).
AQA FP2 2011 January Q5
8 marks Standard +0.8
5
  1. Given that \(u = \sqrt { 1 - x ^ { 2 } }\), find \(\frac { \mathrm { d } u } { \mathrm {~d} x }\).
  2. Use integration by parts to show that $$\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 2 } } \sin ^ { - 1 } x \mathrm {~d} x = a \sqrt { 3 } \pi + b$$ where \(a\) and \(b\) are rational numbers.
AQA FP2 2007 June Q4
7 marks Standard +0.3
4
  1. Differentiate \(x \tan ^ { - 1 } x\) with respect to \(x\).
  2. Show that $$\int _ { 0 } ^ { 1 } \tan ^ { - 1 } x \mathrm {~d} x = \frac { \pi } { 4 } - \ln \sqrt { 2 }$$ (5 marks)