Integrating powers and standard functions Problem set A Worked solutions

Find \(\int x^{\frac{5}{3}}\,dx\).

• Add 1 to the power: \(\frac{5}{3}+1=\frac{8}{3}\).
• Divide by the new power: \(1\div\frac{8}{3}=\frac{3}{8}\).
• \(\int x^{\frac{5}{3}}\,dx=\frac{3}{8}x^{\frac{8}{3}} + c\)

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

• Add 1 to the power: \(-\frac{4}{3}+1=-\frac{1}{3}\).
• Divide by the new power: \(1\div(-\frac{1}{3})=-3\).
• \(\int x^{-\frac{4}{3}}\,dx=-3x^{-\frac{1}{3}} + c\)

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

• Add 1 to the power: \(\frac{1}{3}+1=\frac{4}{3}\).
• Divide by the new power: \(9\div\frac{4}{3}=\frac{27}{4}\).
• \(\int 9x^{\frac{1}{3}}\,dx=\frac{27}{4}x^{\frac{4}{3}} + c\)

Find \(\int \frac{3}{5}\sqrt[3]{x}\,dx\).

• Write the root as a power: \(\sqrt[3]{x}=x^{\frac{1}{3}}\).
• Add 1 to the power: \(\frac{1}{3}+1=\frac{4}{3}\).
• Divide by the new power: \(\frac{3}{5}\div\frac{4}{3}=\frac{9}{20}\).
• \(\int \frac{3}{5}\sqrt[3]{x}\,dx=\frac{9}{20}x^{\frac{4}{3}} + c\)

Find \(\int \frac{6}{\sqrt[3]{x}}\,dx\).

• Write as a negative power: \(\frac{6}{\sqrt[3]{x}}=6x^{-\frac{1}{3}}\).
• Add 1 to the power: \(-\frac{1}{3}+1=\frac{2}{3}\).
• Divide by the new power: \(6\div\frac{2}{3}=9\).
• \(\int \frac{6}{\sqrt[3]{x}}\,dx=9x^{\frac{2}{3}} + c\)

Find y in terms of x.

1. (a) \(\frac{dy}{dx}=5\cos x\)

   • The integral of \(\cos x\) is \(\sin x\).
   • \(y=5\sin x+c\)

2. (b) \(\frac{dy}{dx}=3\sin x\)

   • The integral of \(\sin x\) is \(-\cos x\).
   • \(y=-3\cos x+c\)

A curve has \(\frac{dy}{dx}=6e^x\) and passes through \((0,\,10)\). Find its equation.

• \(y=6e^x+c\)
• \(10=6e^0+c=6+c, \quad c=4\)
• \(y=6e^x+4\)

A curve has \(\frac{dy}{dx}=\frac{8}{x}\) for \(x>0\) and passes through \((1,\,3)\). Find its equation.

• \(y=8\ln|x|+c\)
• \(3=8\ln 1+c=c\), since \(\ln 1=0\)
• \(y=8\ln x+3\)

Find y, given that \(\frac{dy}{dx}=4\sin x-e^x+5\).

• \(\int 4\sin x\,dx=-4\cos x\): the integral of \(\sin x\) is \(-\cos x\).
• \(\int -e^x\,dx=-e^x\), \(\int 5\,dx=5x\).
• \(y=-4\cos x-e^x+5x+c\). Add one c for the whole sum.

The height h metres of a young tree changes at the rate \(\frac{dh}{dt}=\frac{3}{\sqrt{t}}\), where \(t\) is the time in years and \(t>0\). When \(t=1\), \(h=2\) m.

1. (a) Find an expression for \(h\) in terms of \(t\).

   • Write as a power: \(\frac{3}{\sqrt{t}}=3t^{-\frac{1}{2}}\).
   • \(h=\int 3t^{-\frac{1}{2}}\,dt=6t^{\frac{1}{2}}+c\)
   • Put in the point: \(2=6(1)^{\frac{1}{2}}+c\), so \(c=-4\).

2. (b) Find \(h\) when \(t=9\), to 3 significant figures.

   • \(h=6(9)^{\frac{1}{2}}-4=14\) m

A ball rolls through thick sand. Its velocity is \(\frac{ds}{dt}=12e^{-2t}\) m/s, where \(s\) metres is the distance from the start and \(t\) is the time in seconds. When \(t=0\), \(s=0\).

1. (a) Find \(s\) in terms of \(t\).

   • \(s=\int 12e^{-2t}\,dt=-6e^{-2t}+c\): divide by \(-2\), the number in front of t.
   • Put in the point: \(0=-6e^0+c\), so \(c=6\).

2. (b) Find \(s\) when \(t=1.5\), to 3 significant figures.

   • \(s=-6e^{-2\times 1.5}+6=5.70\) m

A curve has \(\frac{dy}{dx}=5x^{\frac{3}{2}}-\frac{4}{x}+\frac{2}{\sqrt{x}}\) for \(x>0\), and passes through \((4,\,20)\). Find the equation of the curve, with the constant to 3 significant figures.

• Write \(\frac{2}{\sqrt{x}}\) as \(2x^{-\frac{1}{2}}\), then integrate each term.
• \(y=2x^{\frac{5}{2}}-4\ln x+4x^{\frac{1}{2}}+c\)
• Put in the point: \(20=2(4)^{\frac{5}{2}}-4\ln 4+4(4)^{\frac{1}{2}}+c\).
• \(c=-46.5\), so \(y=2x^{\frac{5}{2}}-4\ln x+4x^{\frac{1}{2}}-46.5\)

Find \(\int \frac{(\sqrt{x}+3)^2}{x}\,dx\).

• Expand the top: \((\sqrt{x}+3)^2=x+6\sqrt{x}+9\).
• Divide every term by x: \(1+6x^{-\frac12}+\frac{9}{x}\).
• Integrate each term: \(x+12x^{\frac12}+9\ln|x|+c\).