Expanding, splitting fractions and boundary conditions Problem set A

Expand the brackets, then integrate.

1. (a) Find \(\int x^{3}(2x+7)\,dx\).
2. (b) Find \(\int (x+5)(x-4)\,dx\).

Expand the squared bracket, then integrate.

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

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

1. (a) Find \(\int \frac{7x^{4}-2x^{2}}{x^{2}}\,dx\).
2. (b) Find \(\int \frac{3x+8}{4x^{3}}\,dx\).

Find \(\int\left(\frac{5}{2t^{3}}-\frac{3}{t^{6}}\right)dt\).

Find an expression for y in terms of x.

1. (a) \(\frac{dy}{dx}=8x^{3}-3\) and \(y=10\) when \(x=1\).
2. (b) \(\frac{dy}{dx}=9-\frac{4}{x^{2}}\) and \(y=-2\) when \(x=2\).

A curve has gradient \(\frac{dy}{dx}=\frac{10}{x^{2}}-6x\) and passes through the point \((2,3)\). Find the equation of the curve.

A skateboarder's velocity along a straight path is \(v=\frac{ds}{dt}=3t^{2}-8t+5\) m/s, where s is the displacement in metres from the start and t the time in seconds. At \(t=0\) she is at the start, so \(s=0\).

1. (a) Find an expression for s in terms of t.
2. (b) Find how far she has gone when \(t=3\).

The gradient of a ski slope is \(\frac{dy}{dx}=2x-\frac{12}{x^{2}}\), where y is the height in metres and x the horizontal distance in hundreds of metres. The slope passes through the point \((3,20)\).

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

A curve has gradient \(\frac{dy}{dx}=(x+2)(x-5)\) and passes through the point \((3,0)\). Find the equation of the curve.

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