4.08c Improper integrals: infinite limits or discontinuous integrands

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int_1^{\infty} \frac{1}{\cosh u} du,$$ giving your answer in an exact form. [7 marks]

In this question you must show detailed reasoning. Find \(\int_{2}^{\infty} \frac{1}{4+x^2} \, dx\). [4]

In this question you must show detailed reasoning.

1. The curves with equations $$y = \frac{3}{4}\sinh x \text{ and } y = \tanh x + \frac{1}{5}$$ intersect at just one point \(P\)
   1. Use algebra to show that the \(x\) coordinate of \(P\) satisfies the equation $$15e^{4x} - 48e^{3x} + 32e^x - 15 = 0$$ [3]
   2. Show that \(e^x = 3\) is a solution of this equation. [1]
   3. Hence state the exact coordinates of \(P\). [1]
2. Show that $$\int_{-4}^{0} \frac{e^x}{x^2} dx = e^{-\frac{1}{4}}$$ [4]

In this question you must show detailed reasoning. Find \(\int_{-1}^{11} \frac{1}{\sqrt{x^2 + 6x + 13}} dx\) giving your answer in the form \(\ln(p + q\sqrt{2})\) where \(p\) and \(q\) are integers to be determined. [7]

In this question you must show detailed reasoning. Show that \(\int_5^{\infty} (x-1)^{-2} dx = 1\). [5]

In this question you must show detailed reasoning.

1. Find \(\int_{-\frac{3\pi}{4}}^{\frac{3\pi}{4}} 2\tan x \, dx\) giving your answer in the form \(\ln p\). [3]
2. Show that \(\int_0^{\frac{3\pi}{4}} 2\tan x \, dx\) is undefined explaining your reasoning. [2]

A curve has polar equation \(r = \frac{3}{10}e^{3\theta}\) for \(\theta \geq 0\). The length of the arc of this curve between \(\theta = 0\) and \(\theta = \alpha\) is denoted by \(L(\alpha)\).

1. Show that \(L(\alpha) = \frac{1}{3}(e^{3\alpha} - 1)\). [5]
2. The point \(P\) on the curve corresponding to \(\theta = \beta\) is such that \(L(\beta) = OP\), where \(O\) is the pole. Find the value of \(\beta\). [2]