Expanding, splitting fractions and boundary conditions Problem set B Worked solutions

Expand the brackets, then integrate.

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

   • Expand: \(3x^{3} - 5x^{2}\)
   • Integrate: \(\frac{3}{4}x^{4} - \frac{5}{3}x^{3}+c\)

2. (b) Find \(\int (x-6)(x+2)\,dx\).

   • Expand: \(x^{2} - 4x - 12\)
   • Integrate: \(\frac{1}{3}x^{3} - 2x^{2} - 12x+c\)

Expand the squared bracket, then integrate.

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

   • Expand: \(4x^{2} + x^{-2} - 4\)
   • Integrate: \(\frac{4}{3}x^{3} - 4x - x^{-1}+c\)

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

   • Expand: \(x^{4} + 9x^{-4} - 6\)
   • Integrate: \(\frac{1}{5}x^{5} - 6x - 3x^{-3}+c\)

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

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

   • Split into powers of x: \(9x^{2} - 4\)
   • Integrate: \(3x^{3} - 4x+c\)

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

   • Split into powers of x: \(\frac{5}{2}x^{-3} - 3x^{-4}\)
   • Integrate: \(-\frac{5}{4}x^{-2} + x^{-3}+c\)

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

• Write as powers of t: \(\frac{7}{3}t^{-2} + 4t^{-5}\)
• Integrate: \(-\frac{7}{3}t^{-1} - t^{-4}+c\)

Find an expression for y in terms of x.

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

   • Integrate: \(y=2x^{3} - 5x+c\)
   • Substitute \(x=2\), \(y=3\): \(3=6+c\)
   • \(c=-3\)
   • \(y=2x^{3} - 5x - 3\)

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

   • Integrate: \(y=12x + 9x^{-2}+c\)
   • Substitute \(x=3\), \(y=4\): \(4=37+c\)
   • \(c=-33\)
   • \(y=12x + 9x^{-2} - 33\)

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.

• Write as powers: \(4x + 8x^{-3}\)
• Integrate: \(y=2x^{2} - 4x^{-2}+c\)
• Substitute \(x=2\), \(y=9\): \(9=7+c\)
• \(c=2\)
• \(y=2x^{2} - 4x^{-2} + 2\)

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.

   • Integrate: \(V=\frac{4}{3}t^{3} - 3t^{2} + 7t+c\)
   • Put \(t=0\), \(V=0\): \(0=0+c\), so \(c=0\)

2. (b) Find the volume in the trough when \(t=3\).

   • Put \(t=3\): \(V=\frac{4}{3}(3)^{3} - 3(3)^{2} + 7(3)=30\)

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.

   • Write as powers: \(3x^{2} - 16x^{-3}\)
   • Integrate: \(y=x^{3} + 8x^{-2}+c\)
   • Substitute \(x=2\), \(y=30\): \(30=10+c\)
   • \(c=20\)
   • \(y=x^{3} + 8x^{-2} + 20\)

2. (b) Find the height when \(x=1\).

   • Put \(x=1\): \(y=(1)^{3} + 8(1)^{-2} + 20=29\)

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

• Expand: \(x^{2} + 3x - 4\)
• Integrate: \(y=\frac{1}{3}x^{3} + \frac{3}{2}x^{2} - 4x+c\)
• Substitute \(x=3\), \(y=5\): \(5=\frac{21}{2}+c\)
• \(c=-\frac{11}{2}\)
• \(y=\frac{1}{3}x^{3} + \frac{3}{2}x^{2} - 4x - \frac{11}{2}\)

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

• Split into powers of x: \(9x^{-2} + 6x^{-4}\)
• Integrate: \(y=-9x^{-1} - 2x^{-3}+c\)
• Substitute \(x=1\), \(y=1\): \(1=-11+c\)
• \(c=12\)
• \(y=-9x^{-1} - 2x^{-3} + 12\)