\( \int \frac { 1 - \sin x } { 1 + \cos x } d x \)

Установите сходимость или расходимость указанных рядов с помощью интегрального признака Коши. \( \begin{array}{lll}\text { 1) } \frac{1}{2 \ln 2}+\frac{1}{3 \ln 3}+\ldots+\frac{1}{(n+1) \ln (n+1)}+\ldots \\ \text { 2) } \frac{1}{2 \ln ^{2} 2}+\frac{1}{3 \ln ^{2} 3}+\ldots+\frac{1}{(n+1) \ln ^{2}(n+1)}+\ldots ; \\ \sum_{n=1}^{\infty} \frac{\ln (n+1)}{(n+1)^{2}} ; & \text { 4) } \sum_{n=2}^{\infty} \frac{1}{\sqrt{n}} \cdot \ln \frac{n+1}{n-1} ; & \text { 5) } \sum_{n=1}^{\infty} \frac{1}{(n+1) \ln (n+1) \ln \ln (n+1)}\end{array} \)

Find \( \frac{d^{2} y}{d x^{2}} \) given \( x^{3}-3 x y+y^{3}=1 \)

(iv) \( f(x)=\left\{\begin{array}{cc}x \sin \frac{1}{x} & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array} \quad\right. \) at \( x=0 \)

\( \lim _ { x \rightarrow a } ( a - x ) \tan ( \frac { x x } { 20 } ) \)

Express \( \int_{0}^{2} x^{m}\left(1-x^{n}\right)^{p} d x \) in terms of Beta and Gamma function were enaluate \( \int_{0}^{1} x^{2}\left(1-x^{2}\right)^{4} d x \)