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Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Let $f(x)$ be a continuous increasing function in the interval $a\leq x\leq b$ where $0 < a < b$ and $0\leq f(a) < f(b)$. Prove the following formula: $$\int_{f(a)}^{f(b)} f^{-1}(t)dt + \int_a^bf(x)dx = bf(b) - af(a).$$

Find the value of $\int _1^4 \sqrt t dt$, firstly by evaluating the integral directly, and secondly by using the formula above with $f(x)=x^2$.

Use the formula to evaluate $\int_0^1\sin^{-1}t dt.$
What other functions can you integrate more easily using this formula than by other means? 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?