Integration using chain rule reversal

Questions requiring integration of composite functions by recognizing them as derivatives of chain rule expressions, or using substitution.

3 questions · Moderate -0.1

1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
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Edexcel P3 2023 June Q3
7 marks Standard +0.3
  1. (i) Find \(\frac { \mathrm { d } } { \mathrm { d } x } \ln \left( \sin ^ { 2 } 3 x \right)\) writing your answer in simplest form.
    (ii) (a) Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( 3 x ^ { 2 } - 4 \right) ^ { 6 }\) (b) Hence show that
$$\int _ { 0 } ^ { \sqrt { 2 } } x \left( 3 x ^ { 2 } - 4 \right) ^ { 5 } \mathrm {~d} x = R$$ where \(R\) is an integer to be found.
(Solutions relying on calculator technology are not acceptable.)
OCR MEI Paper 2 2019 June Q2
4 marks Moderate -0.8
2 Given that \(y = \left( x ^ { 2 } + 5 \right) ^ { 12 }\),
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find \(\int 48 x \left( x ^ { 2 } + 5 \right) ^ { 11 } \mathrm {~d} x\).
Edexcel C2 Q25
11 marks Standard +0.3
Given that \(f'(x) = (2x^3 - 3x^{-2})^2 + 5\), \(x > 0\),
  1. Find, to 3 significant figures, the value of \(x\) for which \(f'(x) = 5\). [3]
  2. Show that \(f'(x)\) may be written in the form \(Ax^6 + \frac{B}{x^4} + C\), where \(A\), \(B\) and \(C\) are constants to be found. [3]
  3. Hence evaluate \(\int_1^2 f'(x) \, dx\). [5]