Square root substitution: indefinite integral

Use a substitution involving square roots (u = √x, u = 1-√x, u = 1+√x, u² = ax+b) to find an indefinite integral, with no limits of integration.

6 questions · Standard +0.2

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OCR C4 Q2
6 marks Standard +0.3
  1. Use the substitution \(u = 1 - x ^ { \frac { 1 } { 2 } }\) to find
$$\int \frac { 1 } { 1 - x ^ { \frac { 1 } { 2 } } } \mathrm {~d} x$$
OCR H240/02 2019 June Q1
11 marks Moderate -0.3
1
  1. Differentiate the following.
    1. \(\frac { x ^ { 2 } } { 2 x + 1 }\)
    2. \(\tan \left( x ^ { 2 } - 3 x \right)\)
  2. Use the substitution \(u = \sqrt { x } - 1\) to integrate \(\frac { 1 } { \sqrt { x } - 1 }\).
  3. Integrate \(\frac { x - 2 } { 2 x ^ { 2 } - 8 x - 1 }\).
OCR MEI Paper 2 2021 November Q16
8 marks Standard +0.8
16 In this question you must show detailed reasoning.
Find \(\int \frac { x } { 1 + \sqrt { x } } d x\). END OF QUESTION PAPER
AQA C3 2008 January Q5
9 marks Standard +0.3
5
    1. Given that \(y = 2 x ^ { 2 } - 8 x + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence, or otherwise, find $$\int _ { 4 } ^ { 6 } \frac { x - 2 } { 2 x ^ { 2 } - 8 x + 3 } d x$$ giving your answer in the form \(k \ln 3\), where \(k\) is a rational number.
  2. Use the substitution \(u = 3 x - 1\) to find \(\int x \sqrt { 3 x - 1 } \mathrm {~d} x\), giving your answer in terms of \(x\).
OCR MEI Paper 3 2019 June Q11
6 marks Standard +0.3
11 By using the substitution \(u = 1 + \sqrt { x }\), find \(\int \frac { x } { 1 + \sqrt { x } } \mathrm {~d} x\). Answer all the questions.
Pre-U Pre-U 9794/1 2010 June Q4
5 marks Moderate -0.3
Using the substitution \(u = 1 + \sqrt{x}\), or otherwise, find \(\int \frac{1}{1 + \sqrt{x}} dx\) giving your answer in terms of \(x\). [5]