Take a look at this diagram.

Graphs of y=f(x), y=10-3f(x) and y=3f(3x-45) drawn on the same axes. The intercepts of y=3f(3x-45) are 15, 25 and 30. The largest intercepts of y=f(x) is also at x=15. A region between y=f(x) and the x axis has been shaded blue. A region between y=10-3f(x) and the x axis has been shaded orange.

The area of the blue region is \(53\) and the area of the orange region is \(125\).

Use this to explain why \[\int_{15}^{30}3f(2x-45)\,dx=67.\]

Now try to evaluate each of the following integrals.

  1. \(\int_{-4}^{-2}\sqrt{x+5}\,dx\)

  2. \(\int_{0.2}^{1.3}e^{2x}\,dx\)

Give some examples of other functions that you can integrate in this way.