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Age 16 to 18 Challenge Level:

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?