1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

This is the graph of \(y = \frac{5}{4x-3} - \frac{3}{2}\) \includegraphics{figure_6} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a,b,c \in Q\) [6]

In this question you must use detailed reasoning.

1. Show that \(\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1+\sin 2x}{-\cos 2x} dx = \ln(\sqrt{a} + b)\), where \(a\) and \(b\) are integers to be determined. [6]
2. Show that \(\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1+\sin 2x}{-\cos 2x} dx\) is undefined, explaining your reasoning clearly. [2]

1. Determine the values of \(p\) and \(q\) for which $$x^2 - 6x + 10 \equiv (x - p)^2 + q.$$ [2]

1. Use the substitution \(x - p = \tan u\), where \(p\) takes the value found in part (i), to evaluate $$\int_3^4 \frac{1}{x^2 - 6x + 10} \, dx.$$ [3]

1. Determine the value of $$\int_3^4 \frac{x}{x^2 - 6x + 10} \, dx,$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants to be determined. [5]

Find the exact value of \(\int_0^1 (e^x - x) dx\). [4]

Given that $$\int_0^{\frac{\pi}{2}} (1 + \tan\left[\frac{1}{2}x\right])^2 \, dx = a + \ln b$$ find the value of \(a\) and the value of \(b\). [Total 7 marks]