Question

Consider the function

$F\left(x\right)={\int }_3^{x^2}\sin tdt$

Write the above function as composition of two functions and determine the inside and outside function.

Short answer

$A\left(x\right)={\int }_3^{x^2}\sin tdt\mathrm{is}\mathrm{the}\mathrm{outside}\mathrm{function}\mathrm{and}g\left(x\right)=x^2\mathrm{is}\mathrm{the}\mathrm{inside}\mathrm{function}$

Step-by-step solution

Step 1. Given

The given function is$F\left(x\right)={\int }_3^{x^2}\sin tdt$

Step 2. Determining the two function

$$\begin{gathered} \mathrm{The}\mathrm{function}\mathrm{can}\mathrm{be}\mathrm{written}\mathrm{as} \\ F\left(x\right)=A\left(g\right(x\left)\right) \\ whereA\left(x\right)={\int }_3^{x^2}\sin tdtandg\left(x\right)=x^2 \\ \mathrm{Here}\mathrm{A}\left(\mathrm{x}\right)={\int }_3^{{\mathrm{x}}^2}\sin \mathrm{t}\mathrm{dt}\mathrm{is}\mathrm{the}\mathrm{outside}\mathrm{function}\mathrm{and}\mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2\mathrm{is}\mathrm{the}\mathrm{inside}\mathrm{function} \end{gathered}$$