1.08b Integrate x^n: where n != -1 and sums

453 questions

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Edexcel C1 Q3
5 marks Easy -1.3
\(y = 7 + 10x^{\frac{1}{3}}\).
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find \(\int y \, dx\). [3]
Edexcel C1 Q6
7 marks Moderate -0.8
$$\frac{dy}{dx} = 5 + \frac{1}{x^2}.$$
  1. Use integration to find \(y\) in terms of \(x\). [3]
  2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\). [4]
Edexcel C1 Q10
13 marks Moderate -0.3
The curve \(C\) has the equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 8x - \frac{2}{x^3}, \quad x \neq 0,$$ and that the point \(P(1, 1)\) lies on \(C\),
  1. find an equation for the tangent to \(C\) at \(P\) in the form \(y = mx + c\), [3]
  2. find an equation for \(C\), [5]
  3. find the \(x\)-coordinates of the points where \(C\) meets the \(x\)-axis, giving your answers in the form \(k\sqrt{2}\). [5]
Edexcel C1 Q2
4 marks Easy -1.2
Find $$\int \left( 3x^2 + \frac{1}{2x^2} \right) dx.$$ [4]
Edexcel C1 Q8
9 marks Moderate -0.3
Given that $$\frac{dy}{dx} = \frac{x^3 - 4}{x^2}, \quad x \neq 0,$$
  1. find \(\frac{d^2y}{dx^2}\). [3]
Given also that \(y = 0\) when \(x = -1\),
  1. find the value of \(y\) when \(x = 2\). [6]
Edexcel C1 Q5
8 marks Moderate -0.8
\(\text{f}(x) = (2 - \sqrt{x})^2, \quad x > 0\).
  1. Solve the equation \(\text{f}(x) = 0\). [2]
  2. Find \(\text{f}(3)\), giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [2]
  3. Find $$\int \text{f}(x) \, dx.$$ [4]
Edexcel C1 Q7
8 marks Moderate -0.3
Given that $$\text{f}'(x) = 5 + \frac{4}{x^2}, \quad x \neq 0,$$
  1. find an expression for \(\text{f}(x)\). [3]
Given also that $$\text{f}(2) = 2\text{f}(1),$$
  1. find \(\text{f}(4)\). [5]
AQA C2 2009 June Q2
8 marks Moderate -0.8
  1. Write down the value of \(n\) given that \(\frac{1}{x^3} = x^n\). [1]
  2. Expand \(\left(1 + \frac{3}{x^2}\right)^2\). [2]
  3. Hence find \(\int \left(1 + \frac{3}{x^2}\right)^2 dx\). [3]
  4. Hence find the exact value of \(\int_1^3 \left(1 + \frac{3}{x^2}\right)^2 dx\). [2]
Edexcel C2 Q3
7 marks Moderate -0.8
  1. Expand (2√x + 3)². [2]
  2. Hence evaluate $$\int_1^{2^2} (2\sqrt{x} + 3)^2 \, dx$$, giving your answer in the form a + b√2, where a and b are integers. [5]
Edexcel C2 Q4
11 marks Standard +0.3
Given that \(\text{f}(x) = (2x^{\frac{1}{3}} - 3x^{-\frac{1}{2}})^2 + 5\), \(x > 0\),
  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 \(Ax^{\frac{2}{3}} + \frac{B}{x} + C\), where A, B and C are constants to be found. [3]
  3. Hence evaluate \(\int_1^2 \text{f}(x) \, \text{dx}\). [5]
Edexcel C2 Q2
10 marks Moderate -0.8
  1. Differentiate with respect to x $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5 marks]
  2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5 marks]
Edexcel C2 Q2
7 marks Moderate -0.8
  1. Expand \((2\sqrt{x} + 3)^2\). [2]
  2. Hence evaluate \(\int_1^2 (2\sqrt{x} + 3)^2 \, dx\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers. [5]
Edexcel C2 Q5
10 marks Moderate -0.8
  1. Differentiate \(2x^2 + \sqrt{x} + \frac{x^2 + 2x}{x^2}\) with respect to \(x\) [5]
  2. Evaluate \(\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx\). [5]
OCR C2 Q3
7 marks Moderate -0.8
  1. Find \(\int (2x + 1)(x + 3) \, dx\). [4]
  2. Evaluate \(\int_0^9 \frac{1}{\sqrt{x}} \, dx\). [3]
OCR C2 Q6
8 marks Moderate -0.8
  1. Find the binomial expansion of \(\left(x^2 + \frac{1}{x}\right)^3\), simplifying the terms. [4]
  2. Hence find \(\int \left(x^2 + \frac{1}{x}\right)^3 dx\). [4]
OCR C2 2007 January Q3
5 marks Easy -1.2
  1. Find \(\int (4x - 5) dx\). [2]
  2. The gradient of a curve is given by \(\frac{dy}{dx} = 4x - 5\). The curve passes through the point \((3, 7)\). Find the equation of the curve. [3]
OCR C2 2007 January Q10
10 marks Standard +0.3
\includegraphics{figure_10} The diagram shows the graph of \(y = 1 - 3x^{-\frac{1}{2}}\).
  1. Verify that the curve intersects the \(x\)-axis at \((9, 0)\). [1]
  2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\)). Given that the area of the shaded region is 4 square units, find the value of \(a\). [9]
OCR C2 Specimen Q2
6 marks Easy -1.2
  1. Find \(\int \frac{1}{x^2} dx\). [3]
  2. The gradient of a curve is given by \(\frac{dy}{dx} = \frac{1}{x^2}\). Find the equation of the curve, given that it passes through the point \((1, 3)\). [3]
OCR MEI C2 2010 January Q1
3 marks Easy -1.2
Find \(\int \left(x - \frac{3}{x^2}\right) dx\). [3]
OCR MEI C2 2013 January Q1
3 marks Easy -1.8
Find \(\int 30x^2 dx\). [3]
OCR MEI C2 2006 June Q4
5 marks Moderate -0.8
Find \(\int_1^2 \left( x^4 - \frac{3}{x^2} + 1 \right) dx\), showing your working. [5]
OCR MEI C2 2008 June Q12
12 marks Moderate -0.8
\includegraphics{figure_12} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
  1. Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough. [5]
  2. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8x^3 - 3x^2 - 0.5x - 0.15\), for \(0 \leq x \leq 0.5\). Calculate \(\int_0^{0.5} (8x^3 - 3x^2 - 0.5x - 0.15) \, \text{d}x\) and state what this represents. Hence find the volume of water in the trough as given by this model. [7]
OCR MEI C2 2010 June Q5
4 marks Moderate -0.8
Find \(\int_{2}^{5} \left(1 - \frac{6}{x^3}\right) dx\). [4]
OCR MEI C2 2010 June Q6
5 marks Moderate -0.5
The gradient of a curve is \(6x^2 + 12x^{\frac{1}{2}}\). The curve passes through the point \((4, 10)\). Find the equation of the curve. [5]
OCR MEI C2 2013 June Q3
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{18}{x^3} + 2\). The curve passes through the point \((3, 6)\). Find the equation of the curve. [5]