Integrating f(ax+b) and by inspection Problem set B

Find each integral.

1. (a) \(\int \sqrt{3x+7}\,dx\)
2. (b) \(\int \sqrt[3]{2x-5}\,dx\)

Find \(\int 10(5x-2)^{-\frac{3}{2}}\,dx\).

Find each integral.

1. (a) \(\int \cos(6x+2)\,dx\)
2. (b) \(\int 8\cos(4x-5)\,dx\)

Find each integral.

1. (a) \(\int \sin(5x-1)\,dx\)
2. (b) \(\int 9\sin(2-3x)\,dx\)

Find each integral.

1. (a) \(\int 6e^{3x-2}\,dx\)
2. (b) \(\int 12e^{4-3x}\,dx\)

Find each integral.

1. (a) \(\int \frac{10}{5x-1}\,dx\)
2. (b) \(\int \frac{8}{3-4x}\,dx\)

Find \(\int 10x(x^2+4)^{3}\,dx\).

Find each integral.

1. (a) \(\int 6\cos x\sin^{2}x\,dx\)
2. (b) \(\int \sin x\cos^{5}x\,dx\)

Find each integral.

1. (a) \(\int \frac{6x}{x^2+2}\,dx\)
2. (b) \(\int \frac{4x^{3}}{x^4+9}\,dx\)

Find \(\int 9x^{2}e^{x^3+2}\,dx\).

Find \(\int \frac{15}{\cos^2(5x+1)}\,dx\).

Oil leaks from a tank at the rate \(\frac{dV}{dt}=30e^{-0.3t}\) 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 \(t=6\), to 3 significant figures.

The number of visitors to a website, \(V\), is growing at the rate \(\frac{dV}{dt}=\frac{450}{3t+2}\) per day, where \(t\) is the time in days. When \(t=0\), \(V=500\).

1. (a) Find \(V\) in terms of \(t\).
2. (b) Find \(V\) when \(t=8\), to 3 significant figures.

Find \(\int 6x^2\sqrt{x^3+1}\,dx\).

Given that \(f'(x)=\frac{6\sin x}{3-\cos x}\) and \(f(\pi)=5\), find \(f(x)\), with the constant to 3 significant figures.