Let f(x) be a continuous increasing function in the interval a\leq x\leq b where 0 < a < b and 0\leq f(a) < f(b).
Can you prove the following formula with the help of a sketch?\int_{f(a)}^{f(b)} f^{-1}(t) \,dt + \int_a^bf(x) \,dx = bf(b) - af(a).
Why must f(x) be increasing in the interval a\leq x\leq b?
How could you evaluate a similar integral if f(x) is decreasing?
Once you've proved the formula, find the value of \int _1^4 \sqrt t \,dt, in two different ways; firstly by evaluating the integral directly, and secondly by using the formula above with f(x)=x^2.
Have a go at using the formula to evaluate \int_0^1\sin^{-1}t \,dt.
Can you find other functions that you can integrate more easily using this formula than by other means?