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

453 questions

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Edexcel P1 2018 Specimen Q1
6 marks Easy -1.2
Given that \(y = 4x^3 - \frac{5}{x^2}\), \(x \neq 0\), find in their simplest form
  1. \(\frac{dy}{dx}\). [3]
  2. \(\int y \, dx\) [3]
Edexcel P1 2018 Specimen Q7
5 marks Moderate -0.8
A curve with equation \(y = \text{f}(x)\) passes through the point \((4, 25)\) Given that $$\text{f}'(x) = \frac{3}{8}x^2 - 10x^{-\frac{1}{2}} + 1, \quad x > 0$$ find \(\text{f}(x)\), simplifying each term. [5]
Edexcel C1 Q2
8 marks Easy -1.8
  1. Given that \(y = 5x^3 + 7x + 3\), find
    1. \(\frac{dy}{dx}\), [3]
    2. \(\frac{d^2y}{dx^2}\). [1]
  2. Find \(\int \left(1 + 3\sqrt{x} - \frac{1}{x^2}\right) dx\). [4]
Edexcel C1 Q2
5 marks Easy -1.2
Given that \(y = 6x - \frac{4}{x^2}\), \(x \neq 0\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find \(\int y \, dx\). [3]
Edexcel C1 Q7
8 marks Moderate -0.8
  1. Show that \(\frac{(3 - \sqrt{x})^2}{\sqrt{x}}\) can be written as \(9x^{-\frac{1}{2}} - 6 + x^{\frac{1}{2}}\). [2]
Given that \(\frac{dy}{dx} = \frac{(3 - \sqrt{x})^2}{\sqrt{x}}\), \(x > 0\), and that \(y = \frac{2}{3}\) at \(x = 1\),
  1. find \(y\) in terms of \(x\). [6]
Edexcel C1 Q4
5 marks Easy -1.2
Given that \(y = 2x^2 - \frac{6}{x}\), \(x \neq 0\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find \(\int y \, dx\). [3]
Edexcel C1 Q8
7 marks Moderate -0.3
The curve with equation \(y = f(x)\) passes through the point \((1, 6)\). Given that $$f'(x) = 3 + \frac{5x^2 + 2}{x^4}, \quad x > 0,$$ find \(f(x)\) and simplify your answer. [7]
Edexcel C1 Q1
4 marks Easy -1.2
Find \(\int (6x^2 + 2x + x^{-2}) \, dx\), giving each term in its simplest form. [4]
Edexcel C1 Q6
5 marks Moderate -0.8
  1. Show that \((4 + 3\sqrt{x})^3\) can be written as \(16 + k\sqrt{x} + 9x\), where \(k\) is a constant to be found. [2]
  2. Find \(\int (4 + 3\sqrt{x})^3 \, dx\). [3]
Edexcel C1 Q20
14 marks Moderate -0.3
The curve \(C\) has equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 3x^2 - 20x + 29$$ and that \(C\) passes through the point \(P(2, 6)\),
  1. find \(y\) in terms of \(x\). [4]
  2. Verify that \(C\) passes through the point \((4, 0)\). [2]
  3. Find an equation of the tangent to \(C\) at \(P\). [3]
The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
  1. Calculate the exact \(x\)-coordinate of \(Q\). [5]
Edexcel C1 Q21
5 marks Easy -1.3
\(y = 7 + 10x^{\frac{3}{2}}\).
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find \(\int y \, dx\). [3]
Edexcel C1 Q24
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 Q36
9 marks Moderate -0.8
The curve \(C\) with equation \(y = f(x)\) is such that $$\frac{dy}{dx} = 3\sqrt{x} + \frac{12}{\sqrt{x}}, \quad x > 0.$$
  1. Show that, when \(x = 8\), the exact value of \(\frac{dy}{dx}\) is \(9\sqrt{2}\). [3]
The curve \(C\) passes through the point \((4, 30)\).
  1. Using integration, find \(f(x)\). [6]
Edexcel C1 Specimen Q2
4 marks Easy -1.2
Find \(\int 5x + 3\sqrt{x} \, dx\) [4]
Edexcel C2 Q1
4 marks Moderate -0.3
Evaluate \(\int_0^1 \frac{1}{\sqrt{x}} \, dx\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers. [4]
Edexcel C2 Q2
5 marks Easy -1.2
  1. Find \(\int \left( 3 + 4x^3 - \frac{2}{x^2} \right) dx\). [3]
  2. Hence evaluate \(\int_1^2 \left( 3 + 4x^3 - \frac{2}{x^2} \right) dx\). [2]
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]
Edexcel C2 Q31
10 marks Moderate -0.8
  1. Differentiate with respect to \(x\) $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5]
  2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5]
Edexcel C3 Q1
10 marks Moderate -0.3
The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that $$f'(x) = x^2 - 2 + \frac{1}{x^2}.$$
  1. Find the value of \(f''(x)\) at \(x = 4\). [3]
  2. Given that \(f(3) = 0\), find \(f(x)\). [4]
  3. Prove that \(f\) is an increasing function. [3]
Edexcel M2 2014 January Q9
12 marks Moderate -0.3
A curve with equation \(y = f(x)\) passes through the point \((3, 6)\). Given that $$f'(x) = (x - 2)(3x + 4)$$
  1. use integration to find \(f(x)\). Give your answer as a polynomial in its simplest form. [5]
  2. Show that \(f(x) = (x - 2)^2(x + p)\), where \(p\) is a positive constant. State the value of \(p\). [3]
  3. Sketch the graph of \(y = f(x)\), showing the coordinates of any points where the curve touches or crosses the coordinate axes. [4]
Edexcel M2 2002 June Q2
8 marks Standard +0.3
A particle \(P\) moves in a straight line so that, at time \(t\) seconds, its acceleration \(a\) m s\(^{-2}\) is given by $$a = \begin{cases} 4t - t^2, & 0 \leq t \leq 3, \\ \frac{27}{t^2}, & t > 3. \end{cases}$$ At \(t = 0\), \(P\) is at rest. Find the speed of \(P\) when
  1. \(t = 3\), [3]
  2. \(t = 6\). [5]
Edexcel M2 2003 June Q1
5 marks Moderate -0.3
A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\) in the direction of \(x\) increasing, where \(v = 6t - 2t^2\). When \(t = 0\), \(P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest after leaving \(O\). [5]
Edexcel M2 2006 June Q1
6 marks Moderate -0.8
A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, its acceleration is \((5 - 2t)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. When \(t = 0\), its velocity is 6 m s\(^{-1}\) measured in the direction of \(x\) increasing. Find the time when \(P\) is instantaneously at rest in the subsequent motion. [6]
Edexcel C1 Q5
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 Q7
14 marks Moderate -0.3
The curve \(C\) has equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 3x^2 - 20x + 29$$ and that \(C\) passes through the point \(P(2, 6)\),
  1. find \(y\) in terms of \(x\). [4]
  2. Verify that \(C\) passes through the point \((4, 0)\). [2]
  3. Find an equation of the tangent to \(C\) at \(P\). [3]
The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
  1. Calculate the exact \(x\)-coordinate of \(Q\). [5]