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

Find each integral.

1. (a) \(\int \sqrt{5x-2}\,dx\)
2. (b) \(\int \sqrt[3]{4x+3}\,dx\)

Find \(\int 12(3x+1)^{-\frac{4}{3}}\,dx\).

Find each integral.

1. (a) \(\int \cos(4x-1)\,dx\)
2. (b) \(\int 5\cos(2x+3)\,dx\)

Find each integral.

1. (a) \(\int \sin(3x+2)\,dx\)
2. (b) \(\int 4\sin(1-2x)\,dx\)

Find each integral.

1. (a) \(\int 4e^{5x+1}\,dx\)
2. (b) \(\int 10e^{1-2x}\,dx\)

Find each integral.

1. (a) \(\int \frac{6}{2x+5}\,dx\)
2. (b) \(\int \frac{3}{4-3x}\,dx\)

Find \(\int 12x^{3}(x^4-2)^{3}\,dx\).

Find each integral.

1. (a) \(\int 5\cos x\sin^{4}x\,dx\)
2. (b) \(\int \sin x\cos^{2}x\,dx\)

Find each integral.

1. (a) \(\int \frac{4x}{x^2+7}\,dx\)
2. (b) \(\int \frac{3x^{2}}{x^3+2}\,dx\)

Find \(\int 8xe^{x^2-3}\,dx\).

Find \(\int \frac{12}{\cos^2(3x-2)}\,dx\).

A trolley slows down on a rough floor. Its velocity is \(\frac{ds}{dt}=40e^{-0.5t}\) cm/s, where \(s\) cm 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 \(t=4\), to 3 significant figures.

The number of bacteria in a dish, \(N\), grows at the rate \(\frac{dN}{dt}=\frac{600}{2t+5}\) per hour, where \(t\) is the time in hours. When \(t=0\), \(N=1000\).

1. (a) Find \(N\) in terms of \(t\).
2. (b) Find \(N\) when \(t=10\), to 3 significant figures.

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

Given that \(f'(x)=\frac{8\cos x}{2+\sin x}\) and \(f(\frac{\pi}{2})=10\), find \(f(x)\), with the constant to 3 significant figures.