Given that \(\mathrm{f}^{\prime \prime}(x)=(3 x+5)^{-\frac{2}{3}}, \mathrm{f}^{\prime}(1)=6\), and \(\mathrm{f}(1)=20\), find an expression for \(\mathrm{f}(x)\).

The second derivative of a curve is given by

\(\frac{d^2y}{dx^2}=(3x+2)^{-\frac13}.\)

The gradient of the curve is \(4\) at the point \((2,6.2)\).

Find the equation of the curve.

A curve is such that \(\frac{d^2y}{dx^2}=(2x+3)^{-1/2}\). The curve has a gradient of \(5\) at the point where \(x=3\), and passes through the point \(\left(\frac12,-\frac13\right)\).

(i) Find the equation of the curve.

(ii) Find the equation of the normal to the curve at the point where \(x=3\), giving your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.

(i) Find \(\displaystyle\int(7x-10)^{-\frac35}\,dx\).

(ii) Given that \(\displaystyle\int_6^a(7x-10)^{-\frac35}\,dx=\dfrac{25}{14}\), find the exact value of \(a\).

(i) Find \(\dfrac{d}{dx}\left(\dfrac5{3x+2}\right)\).

(ii) Use your answer to part (i) to find \(\displaystyle\int\dfrac{30}{(3x+2)^2}\,dx\).

(iii) Hence evaluate \(\displaystyle\int_1^2\dfrac{30}{(3x+2)^2}\,dx\).