Let $f(x)$ be a continuous increasing function in the interval $a\leqq x\leqq b$ where $0<a<b$ and $0\leqq f(a)<f(b)$. Prove the formula

${\displaystyle {\int }_{f(a)}^{f(b)}{f}^{-1}(t)\mathrm{dt}+{\int }_a^{b}f(x)\mathrm{dx}=\mathrm{bf}(b)-\mathrm{af}(a).}$

By considering the function $f(x)=x^2$ find the value of ${\int }_1^4\sqrt{t}\mathrm{dt}$ in two ways, both by evaluating the integral directly and also by using the formula above.

Use the formula to evaluate ${\int }_0^1{\sin }^{-1}t\mathrm{dt}.$