.) \( \int \frac{\sin \ln x}{r} d x \)

1) \( y=\cos ^{-1}(\sin x) \Rightarrow y^{\prime}=? \) A) \( -\frac{\cos x}{|\cos x|} \) B) \( \frac{\cos x}{|\cos x|} \) C) \( -\frac{\cos x}{\sqrt{1-x^{2}}} \) D) \( \frac{\cos x}{\sqrt{1-x^{2}}} \) E) \( -\frac{\sin x}{\sqrt{1-x^{2}}} \)

b) (i) Consider the infinite geometric series \( \sum_{n=1}^{\infty} 10\left(1-\frac{3 x}{2}\right)^{n} \) for what does a sum to infinity exist? (ii) Find the sum of the seriec if.

25) \( \lim _{x \rightarrow 0} \frac{|x-2|}{x-2}= \) ? \( \begin{array}{lllll}\text { A) }-\infty & \text { B) Limit yoktur } & \text { C) } 1 & \text { D) } \infty & \text { E) }-1\end{array} \)

(b) Prove that \( \left|x_{n}-x_{n+1}^{\infty}\right|=\frac{1}{2^{n-1}} \) for all \( n \in \mathbb{N} \)

2) \( \left.\frac{d}{d x} \cot ^{-1}(a x+b)\right|_{x=-b / a}=? \) A) \( a \) B) \( b \) C) \( -a \) D) \( -b \) E) \( a / b \)