Q2. Find the equation for the curve that is an extremal for the functional \[ \begin{array}{l}f(x)=\int_{0}^{t}\left[t \dot{x}(t)+\dot{x}^{2}(t)\right] d t \\ \text { Boundary Conditions are } x(0)=1, t_{f}=2 \text { and } x(2) \text { is free. }\end{array} \]

\( f ( x ) = \left\{ \begin{array} { l l } { \sin x } & { ; x < r } \\ { m x + b ; } & { x \geq r } \end{array} \right. \)

\( \lim _ { x \rightarrow 0 } \frac { 3 ( x - \operatorname { cts } x ) } { x } \)

(c) \( \int_{6}^{0}(2+5 x) e^{\frac{1}{3} x} d x \)

Calcule \( \int \frac{x^{4} - 5x^{2} + 6}{x-2} \,dx \) aplicando o método de divisão longa.

(b) Solve the following first order differential equations. i) \( \frac{d y}{d x}=4 x^{3}-8 x+7 \) kena tambah 1 ii) \( x \frac{d y}{d x}=5 x^{3}+4 \) iii) \( y^{3} \frac{d y}{d x}=\frac{e^{4 x}+4 x}{y} \)