Indefinite integral with non-linear substitution (algebraic/exponential/logarithmic)

Find an indefinite integral using a specified non-linear substitution such as u = ln x, u = x², u = √x, or u = polynomial, returning an answer in terms of x, where the integrand does not involve trigonometric functions requiring the Weierstrass substitution.

8 questions · Standard +0.8

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OCR C4 2013 June Q6
6 marks Standard +0.3
6 Use the substitution \(u = 1 + \ln x\) to find \(\int \frac { \ln x } { x ( 1 + \ln x ) ^ { 2 } } \mathrm {~d} x\).
OCR C4 2016 June Q6
6 marks Standard +0.3
6 Use the substitution \(u = x ^ { 2 } - 2\) to find \(\int \frac { 6 x ^ { 3 } + 4 x } { \sqrt { x ^ { 2 } - 2 } } \mathrm {~d} x\).
Edexcel PURE 2024 October Q6
Challenging +1.2
  1. Use the substitution \(u = \sqrt { x ^ { 3 } + 1 }\) to show that
$$\int \frac { 9 x ^ { 5 } } { \sqrt { x ^ { 3 } + 1 } } \mathrm {~d} x = 2 \left( x ^ { 3 } + 1 \right) ^ { k } \left( x ^ { 3 } - A \right) + c$$ where \(k\) and \(A\) are constants to be found and \(c\) is an arbitrary constant.
Pre-U Pre-U 9794/1 2015 June Q11
10 marks Standard +0.8
11 Using the substitution \(x = u ^ { 2 } - 1\), or otherwise, show that $$\int \frac { 1 } { 2 x \sqrt { x + 1 } } \mathrm {~d} x = \ln \left( A \sqrt { \frac { \sqrt { x + 1 } - 1 } { \sqrt { x + 1 } + 1 } } \right)$$ where \(A\) is an arbitrary constant and \(x > 0\).
AQA C3 2011 June Q8
5 marks Standard +0.8
Use the substitution \(u = 1 + 2\tan x\) to find $$\int \frac{1}{(1 + 2\tan x)^2 \cos^2 x} \, dx$$ [5]
Edexcel C4 Q2
6 marks Standard +0.3
Use the substitution \(u = 1 - x^2\) to find $$\int \frac{1}{1-x^2} \, dx.$$ [6]
OCR H240/03 2017 Specimen Q7
10 marks Challenging +1.3
  1. Find \(\int 5x^3\sqrt{x^2 + 1} dx\). [5]
  2. Find \(\int \theta \tan^2 \theta d\theta\). You may use the result \(\int \tan \theta d\theta = \ln|\sec \theta| + c\). [5]
Pre-U Pre-U 9794/2 2016 June Q10
10 marks Challenging +1.2
  1. Using the substitution \(u = \frac{1}{x}\), or otherwise, find \(\int \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [4]
  2. Evaluate \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) and \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [3]
  3. Show that, when \(n\) is a positive integer, the integral \(\int_{\frac{1}{(n+1)\pi}}^{\frac{1}{n\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) takes one of the two values found in part (ii), distinguishing between the two cases. [3]