Question

To determine the value for \({g^\prime }(x)\) by using the integral evaluation method.

Short answer

The \({g^\prime }(x)\) value is \({x^2}\) by by the use of the integral evaluation method.

Step-by-step solution

Given data

The integral function is \(g(x) = \int_1^x {{t^2}} dt\).

Concept used of differentiation

Let\(y = f(x)\)be a function of\(x\).

Then, the rate of change of "\(y\)" per unit change in "\(x\)" is given by\(\frac{{dy}}{{dx}}\).

Differentiate the given function

The given integral function is,

\(\begin{aligned}{c}g(x) &= \int_1^x {{t^2}} dt\\ &= \left( {\frac{{{t^3}}}{3}} \right)_1^x\\ &= \frac{{{x^3}}}{3} - \frac{1}{3}\end{aligned}\)

Differentiate with respect to \(X\).

\(\begin{aligned}{c}g(x) &= \frac{{{x^3}}}{3} - \frac{1}{3}{g^\prime }(x)\\ &= \frac{1}{3}\left( {3{x^2}} \right) &= {x^2}\end{aligned}\)

Therefore, get the value of \({g^\prime }(x)\) as \(\underline {{x^2}} \) by by the use of the integral evaluation method.