Trigonometric substitution: show transformation then evaluate

Use a substitution of the form x = a sin θ, x = a cos θ, x = a tan θ, or x = a sec θ to first show the integral transforms to a specific trigonometric form, then evaluate the resulting integral.

4 questions · Standard +0.6

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CAIE P3 2009 November Q6
8 marks Standard +0.3
6
  1. Use the substitution \(x = 2 \tan \theta\) to show that $$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$$
Edexcel P4 2022 June Q5
8 marks Standard +0.8
  1. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.}
  1. Use the substitution \(x = 2 \sin u\) to show that $$\int _ { 0 } ^ { 1 } \frac { 3 x + 2 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x = \int _ { 0 } ^ { p } \left( \frac { 3 } { 2 } \operatorname { secutanu } + \frac { 1 } { 2 } \sec ^ { 2 } u \right) d u$$ where \(p\) is a constant to be found.
  2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 3 x + 2 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$
Edexcel C4 2015 June Q6
8 marks Standard +0.8
\includegraphics{figure_2} Figure 2 shows a sketch of the curve with equation \(y = \sqrt{(3-x)(x+1)}\), \(0 \leqslant x \leqslant 3\) The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis, and the \(y\)-axis.
  1. Use the substitution \(x = 1 + 2\sin\theta\) to show that $$\int_0^3 \sqrt{(3-x)(x+1)} dx = k \int_{-\frac{\pi}{6}}^{\frac{\pi}{2}} \cos^2\theta d\theta$$ where \(k\) is a constant to be determined. [5]
  2. Hence find, by integration, the exact area of \(R\). [3]
OCR C4 2005 June Q4
7 marks Standard +0.3
  1. Show that the substitution \(x = \tan \theta\) transforms \(\int \frac{1}{(1 + x^2)^2} dx\) to \(\int \cos^2 \theta d\theta\). [3]
  2. Hence find the exact value of \(\int_0^1 \frac{1}{(1 + x^2)^2} dx\). [4]