Q. 11 - Calculus

Question

Consider the function

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

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

Step-by-Step Solution

Answer

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

Step 1. Given

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

Step 2. Determining the two function

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