Square root substitution: definite integral

Use a substitution involving square roots (u = √x, u² = x, u = √(x+a), u² = ax+b) to evaluate a definite integral with given limits.

10 questions · Standard +0.6

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OCR C4 Q6
10 marks Standard +0.8
6. (i) Find $$\int \cot ^ { 2 } 2 x \mathrm {~d} x$$ (ii) Use the substitution \(u ^ { 2 } = x + 1\) to evaluate $$\int _ { 0 } ^ { 3 } \frac { x ^ { 2 } } { \sqrt { x + 1 } } \mathrm {~d} x$$
OCR C4 2010 June Q4
7 marks Standard +0.3
4 Use the substitution \(u = \sqrt { x + 2 }\) to find the exact value of $$\int _ { - 1 } ^ { 7 } \frac { x ^ { 2 } } { \sqrt { x + 2 } } \mathrm {~d} x$$
OCR H240/03 2018 June Q5
13 marks Standard +0.3
5
  1. Use the trapezium rule, with two strips of equal width, to show that $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x \approx \frac { 11 } { 4 } - \sqrt { 2 }$$
  2. Use the substitution \(x = u ^ { 2 }\) to find the exact value of $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x$$
  3. Using your answers to parts (i) and (ii), show that $$\ln 2 \approx k + \frac { \sqrt { 2 } } { 4 }$$ where \(k\) is a rational number to be determined.
Edexcel Paper 1 2018 June Q13
7 marks Standard +0.3
  1. Show that
$$\int _ { 0 } ^ { 2 } 2 x \sqrt { x + 2 } \mathrm {~d} x = \frac { 32 } { 15 } ( 2 + \sqrt { 2 } )$$
AQA C3 2013 June Q10
15 marks Standard +0.3
10
    1. By writing \(\ln x\) as \(( \ln x ) \times 1\), use integration by parts to find \(\int \ln x \mathrm {~d} x\).
    2. Find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
      (4 marks)
  2. Use the substitution \(u = \sqrt { x }\) to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$ (7 marks)
Edexcel C4 Q6
12 marks Standard +0.3
6. (a) Find $$\int 2 \sin 3 x \sin 2 x d x$$ (b) Use the substitution \(u ^ { 2 } = x + 1\) to evaluate $$\int _ { 0 } ^ { 3 } \frac { x ^ { 2 } } { \sqrt { x + 1 } } \mathrm {~d} x$$ 6. continued
OCR MEI C3 Q8
18 marks Standard +0.3
Fig. 8 shows the graph of \(y = x\sqrt{1 + x}\). The point P on the curve is on the \(x\)-axis. \includegraphics{figure_8}
  1. Write down the coordinates of P. [1]
  2. Show that \(\frac{dy}{dx} = \frac{3x + 2}{2\sqrt{1 + x}}\). [4]
  3. Hence find the coordinates of the turning point on the curve. What can you say about the gradient of the curve at P? [4]
  4. By using a suitable substitution, show that \(\int_0^0 x\sqrt{1 + x} dx = \int_0^1 \left(u^{\frac{3}{2}} - u^{\frac{1}{2}}\right) du\). Evaluate this integral, giving your answer in an exact form. What does this value represent? [7]
  5. Use your answer to part (ii) to differentiate \(y = x\sqrt{1 + x} \sin 2x\) with respect to \(x\). (You need not simplify your result.) [2]
AQA Paper 1 Specimen Q8
7 marks Challenging +1.2
  1. Given that \(u = 2^x\), write down an expression for \(\frac{du}{dx}\) [1 mark]
  2. Find the exact value of \(\int_0^1 2^x \sqrt{3 + 2^x}\) dx Fully justify your answer. [6 marks]
OCR MEI Paper 2 Specimen Q13
6 marks Challenging +1.2
Evaluate \(\int_0^1 \frac{1}{1 + \sqrt{x}} \, dx\), giving your answer in the form \(a + b \ln c\), where \(a\), \(b\) and \(c\) are integers. [6]
SPS SPS FM Pure 2021 June Q11
7 marks Standard +0.8
  1. Given that \(u = 2^x\), write down an expression for \(\frac{du}{dx}\) [1 mark]
  2. Find the exact value of \(\int_0^1 2^x \sqrt{3 + 2^x} dx\) Fully justify your answer. [6 marks]