Expanding, splitting fractions and boundary conditions Problem set B

Expand the brackets, then integrate.

1. (a) Find \(\int x^{2}(3x-5)\,dx\).
2. (b) Find \(\int (x-6)(x+2)\,dx\).

Expand the squared bracket, then integrate.

1. (a) Find \(\int \left(2x-x^{-1}\right)^{2}\,dx\).
2. (b) Find \(\int \left(x^{2}-3x^{-2}\right)^{2}\,dx\).

Write each fraction as a sum of powers of x, then integrate.

1. (a) Find \(\int \frac{9x^{5}-4x^{3}}{x^{3}}\,dx\).
2. (b) Find \(\int \frac{5x-6}{2x^{4}}\,dx\).

Find \(\int\left(\frac{7}{3t^{2}}+\frac{4}{t^{5}}\right)dt\).

Find an expression for y in terms of x.

1. (a) \(\frac{dy}{dx}=6x^{2}-5\) and \(y=3\) when \(x=2\).
2. (b) \(\frac{dy}{dx}=12-\frac{18}{x^{3}}\) and \(y=4\) when \(x=3\).

A curve has gradient \(\frac{dy}{dx}=\frac{8}{x^{3}}+4x\) and passes through the point \((2,9)\). Find the equation of the curve.

Water runs into a trough at a rate of \(\frac{dV}{dt}=4t^{2}-6t+7\) litres per minute, where V is the volume in litres and t the time in minutes. At \(t=0\) the trough is empty, so \(V=0\).

1. (a) Find an expression for V in terms of t.
2. (b) Find the volume in the trough when \(t=3\).

The gradient of a roller-coaster track is \(\frac{dy}{dx}=3x^{2}-\frac{16}{x^{3}}\), where y is the height in metres and x the horizontal distance in tens of metres. The track passes through the point \((2,30)\).

1. (a) Find the equation of the track.
2. (b) Find the height when \(x=1\).

A curve has gradient \(\frac{dy}{dx}=(x-1)(x+4)\) and passes through the point \((3,5)\). Find the equation of the curve.

Given that \(\frac{dy}{dx}=\frac{9x^{2}+6}{x^{4}}\) and \(y=1\) when \(x=1\), find y in terms of x.