Integrating powers and standard functions Problem set A

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

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

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

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

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

Find y in terms of x.

1. (a) \(\frac{dy}{dx}=5\cos x\)
2. (b) \(\frac{dy}{dx}=3\sin x\)

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

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

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

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

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

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.

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