1.08i Integration by parts

The integrals \(C\) and \(S\) are defined by $$C = \int_0^{3\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{3\pi} e^{3x} \sin 3x \, dx.$$ By considering \(C + iS\) as a single integral, show that $$C = -\frac{1}{3}(2 + 3e^{\pi}),$$ and obtain a similar expression for \(S\). [8] (You may assume that the standard result for \(\int e^{kx} dx\) remains true when \(k\) is a complex constant, so that \(\int e^{(a+ib)x} dx = \frac{1}{a+ib}e^{(a+ib)x}\).)

\includegraphics{figure_5} The diagram shows the curve \(C\) with parametric equations \(x = \frac{3}{t}\), \(y = t^2 e^{-2t}\), where \(t > 0\). The maximum point on \(C\) is denoted by \(P\).

1. Determine the exact coordinates of \(P\). [4] The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 6\).
2. Show that the area of \(R\) is given by $$\int_a^b 3te^{-2t} dt,$$ where \(a\) and \(b\) are constants to be determined. [3]
3. Hence determine the exact area of \(R\). [5]

Given that $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} x \cos x \, dx = a\pi + b$$ find the exact value of \(a\) and the exact value of \(b\). Fully justify your answer. [6 marks]

The curve \(C\) with equation $$y = (x^2 - 8x) \ln x$$ is defined for \(x > 0\) and is shown in the diagram below. \includegraphics{figure_11} The shaded region, \(R\), lies below the \(x\)-axis and is bounded by \(C\) and the \(x\)-axis. Show that the area of \(R\) can be written as $$p + q \ln 2$$ where \(p\) and \(q\) are rational numbers to be found. [10 marks]

\(\int_1^2 x^3 \ln(2x) dx\) can be written in the form \(p\ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\). [5 marks]

Given that \(I_n = \int_0^{\frac{\pi}{2}} \sin^n x \, dx\) \quad \(n \geq 0\) show that \(n I_n = (n-1)I_{n-2}\) \quad \(n \geq 2\) [5 marks]

1. A curve is in the first quadrant. It has parametric equations \(x = \cosh t + \sinh t\), \(y = \cosh t - \sinh t\) where \(t \in \mathbb{R}\). Show that the cartesian equation of the curve is \(xy = 1\). [2]

Fig. 6 shows the curve from part (i). P is a point on the curve. O is the origin. Point A lies on the \(x\)-axis, point B lies on the \(y\)-axis and OAPB is a rectangle. \includegraphics{figure_6}

1. Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer. [4]

In this question you must show detailed reasoning. Show that $$\int_0^{\frac{\pi}{3}} \operatorname{arcsinh} 2x \, dx = \frac{2}{3} \ln 3 - \frac{1}{3}.$$ [8]

Evaluate

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

\(\int_1^2 x^3 \ln(2x) dx\) can be written in the form \(p \ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\). [5 marks]

\includegraphics{figure_1} Figure 1 shows the finite region \(R\), which is bounded by the curve \(y = xe^x\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis. The region \(R\) is rotated through 360 degrees about the \(x\)-axis. Use integration by parts to find an exact value for the volume of the solid generated. [7]

\includegraphics{figure_4} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the x-axis is rotated by \(2\pi\) radians around the x-axis to form a solid of revolution, S. Determine the exact volume of S. [7]

In this question you must show detailed reasoning. \includegraphics{figure_5} The function f is defined for the domain \(x \geqslant 0\) by $$\mathrm{f}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ The diagram shows the curve \(y = \mathrm{f}(x)\).

1. Find the range of f. [6]

1. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm{g}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ Given that g is a one-one function, state the least possible value of \(k\). [1]

1. Find the exact area of the shaded region enclosed by the curve and the \(x\)-axis. [7]

\includegraphics{figure_5} The functions f(x) and g(x) are defined for \(x \geqslant 0\) by \(\text{f}(x) = \frac{x}{x^2 + 3}\) and \(\text{g}(x) = \text{e}^{-2x}\). The diagram shows the curves \(y = \text{f}(x)\) and \(y = \text{g}(x)\). The equation \(\text{f}(x) = \text{g}(x)\) has exactly one real root \(\alpha\).

1. Show that \(\alpha\) satisfies the equation \(\text{h}(x) = 0\), where \(\text{h}(x) = x^2 + 3 - x\text{e}^{2x}\). [2]
2. Hence show that a Newton-Raphson iterative formula for finding \(\alpha\) can be written in the form $$x_{n+1} = \frac{x_n^2(1 - 2\text{e}^{2x_n}) - 3}{2x_n - (1 + 2x_n)\text{e}^{2x_n}}.$$ [5]
3. Use this iterative formula, with a suitable initial value, to find \(\alpha\) correct to 3 decimal places. Show the result of each iteration. [3]
4. In this question you must show detailed reasoning. Find the exact value of \(x\) for which \(\text{fg}(x) = \frac{2}{13}\). [6]