Integrating powers and standard functions Problem set B

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

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

Find \(\int 10x^{\frac{3}{5}}\,dx\).

Find \(\int \frac{5}{8}\sqrt{x}\,dx\).

Find \(\int \frac{12}{\sqrt[4]{x}}\,dx\).

Find y in terms of x.

1. (a) \(\frac{dy}{dx}=9\cos x\)
2. (b) \(\frac{dy}{dx}=4\sin x\)

A curve has \(\frac{dy}{dx}=5e^x\) and passes through \((0,\,2)\). Find its equation.

A curve has \(\frac{dy}{dx}=\frac{5}{x}\) for \(x>0\) and passes through \((1,\,-2)\). Find its equation.

Find y, given that \(\frac{dy}{dx}=2\cos x+6e^x-3\).

The depth L metres of water in a harbour basin changes at the rate \(\frac{dL}{dt}=\frac{2}{\sqrt[3]{t}}\), where \(t\) is the time in hours and \(t>0\). When \(t=1\), \(L=5\) m.

1. (a) Find an expression for \(L\) in terms of \(t\).
2. (b) Find \(L\) when \(t=27\), to 3 significant figures.

Oil leaks from a tank. The rate of leak is \(\frac{dV}{dt}=15e^{-3t}\) litres per minute, where \(V\) litres is the amount leaked and \(t\) is the time in minutes. When \(t=0\), \(V=0\).

1. (a) Find \(V\) in terms of \(t\).
2. (b) Find \(V\) when \(t=1\), to 3 significant figures.

A curve has \(\frac{dy}{dx}=12x^{\frac{1}{3}}-\frac{6}{x}+\frac{3}{\sqrt[3]{x}}\) for \(x>0\), and passes through \((8,\,30)\). Find the equation of the curve, with the constant to 3 significant figures.

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