1.08h Integration by substitution

\includegraphics{figure_6} Figure 1 shows the curve with equation \(y = x\sqrt{1-x}\), \(0 \leq x \leq 1\).

1. Use the substitution \(u^2 = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac{8}{15}\). [8]
2. Find, in terms of \(\pi\), the volume of the solid formed when the region bounded by the curve and the \(x\)-axis is rotated through \(360°\) about the \(x\)-axis. [5]

Use the substitution \(x = 2\tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]

Use the substitution \(u = 1 + \sin x\) to find the value of $$\int_0^{\frac{\pi}{4}} \cos x (1 + \sin x)^3 \, dx.$$ [6]

Use the substitution \(u = 1 - x^2\) to find $$\int \frac{1}{1-x^2} \, dx.$$ [6]

Use the substitution \(x = 2 \tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]

A puck, of mass \(m\) kg, is moving in a straight line across smooth horizontal ice. At time \(t\) seconds, the puck has speed \(v \text{ m s}^{-1}\). As the puck moves, it experiences an air resistance force of magnitude \(0.3mv^3\) newtons, until it comes to rest. No other horizontal forces act on the puck. When \(t = 0\), the speed of the puck is \(8 \text{ m s}^{-1}\). Model the puck as a particle.

1. Show that $$v = (4 - 0.2t)^{\frac{3}{2}}$$ [6 marks]
2. Find the value of \(t\) when the puck comes to rest. [2 marks]
3. Find the distance travelled by the puck as its speed decreases from \(8 \text{ m s}^{-1}\) to zero. [5 marks]

A curve is defined parametrically by $$x = t^3 + 5, \quad y = 6t^2 - 1$$ The arc length between the points where \(t = 0\) and \(t = 3\) on the curve is \(s\).

1. Show that \(s = \int_{0}^{3} 3t\sqrt{t^2 + A} \, \text{d}t\), stating the value of the constant \(A\). [4 marks]
2. Hence show that \(s = 61\). [4 marks]

The arc of the curve with equation \(y = 4 - \ln(1-x^2)\) from \(x = 0\) to \(x = \frac{3}{4}\) has length \(s\).

1. Show that \(s = \int_0^{\frac{3}{4}} \frac{\sqrt{1+x^2}}{1-x^2} \, dx\). [4 marks]
2. Find the value of \(s\), giving your answer in the form \(p + \ln N\), where \(p\) is a rational number and \(N\) is an integer. [6 marks]

A curve has equation $$y = \frac{4x - 3a}{2(x^2 + a^2)},$$ where \(a\) is a positive constant.

1. Explain why the curve has no asymptotes parallel to the \(y\)-axis. [2]
2. Find, in terms of \(a\), the set of values of \(y\) for which there are no points on the curve. [5]
3. Find the exact value of \(\int_a^{2a} \frac{4x - 3a}{2(x^2 + a^2)} dx\), showing that it is independent of \(a\). [5]

By first completing the square in the denominator, find the exact value of $$\int_{\frac{1}{2}}^{\frac{1}{2}} \frac{1}{4x^2 - 4x + 5} dx.$$ [5]

1. Find the general solution of the differential equation $$\frac{dy}{dx} - \frac{y}{x} = \sin 2x,$$ expressing \(y\) in terms of \(x\) in your answer. [6]

In a particular case, it is given that \(y = \frac{2}{\pi}\) when \(x = \frac{1}{4}\pi\).

1. Find the solution of the differential equation in this case. [2]
2. Write down a function to which \(y\) approximates when \(x\) is large and positive. [1]

Use integration by substitution to show that $$\int_{-\frac{3}{4}}^6 x\sqrt{4x + 1} \, dx = \frac{875}{12}$$ Fully justify your answer. [6 marks]

The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain. The curve \(y = f(x)\) intersects the line \(y = x\) at the points P and Q as shown below. \includegraphics{figure_10}

1. State the value of \(x\) which is not in the domain of f. [1 mark]
2. Explain how you know that the function f is many-to-one. [2 marks]
3. 1. Show that the \(x\)-coordinates of P and Q satisfy the equation $$x^2 + 5x - 10 = 0$$ [2 marks]
   2. Hence, find the exact \(x\)-coordinate of P and the exact \(x\)-coordinate of Q. [1 mark]
4. Show that P and Q are stationary points of the curve. Fully justify your answer. [5 marks]
5. Using set notation, state the range of f. [2 marks]

Use the substitution \(u = x^5 + 2\) to show that $$\int_0^1 \frac{x^9}{(x^5 + 2)^3} \, dx = \frac{1}{180}$$ [7 marks]

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]

Evaluate the improper integral \(\int_0^{\infty} \frac{4x - 30}{(x^2 + 5)(3x + 2)} \, dx\), showing the limiting process used. Give your answer as a single term. [8 marks]

Evaluate

1. \(\int_0^2 x^3 \ln x \, dx\). [6]
2. \(\int_0^1 \frac{2+x}{\sqrt{4-x^2}} \, dx\). [6]