Question

Find the value of ${\int }_0^2{f}^{\text{'}}\left(x^2\right)2xdx$

• A. $f\left(4\right)–f\left(0\right)$
• B. $f\left(4\right)+f\left(0\right)$
• C. $f\left(0\right)–f\left(4\right)$
• D. None of these

Answer 1

The correct option is A

$f\left(4\right)–f\left(0\right)$

An explanation for the correct option:

Step 1: Convert the integrand to $\mathbf{f}\mathbf{\left(}\mathit{m}\mathbf{\right)}\mathbf{dm}$ using substitution

We need to find the value of ${\int }_0^2{f}^{\text{'}}\left(x^2\right)2xdx$

Let, $x^2=m$

Differentiation w.r.to $x.$

$\Rightarrow 2xdx=dm$

Step 2: Convert the limits of integration to the corresponding values of $m.$

Also, for $x=0$,

$$\begin{gathered} \Rightarrow m=0^2 \\ \Rightarrow m=0 \end{gathered}$$

and for $x=2$,

$$\begin{gathered} \Rightarrow m=2^2 \\ \Rightarrow m=4 \end{gathered}$$

Step 3: Evaluate the integral in terms of $m$.

So given integration would be,

$$\begin{gathered} ={\int }_0^4{f}^{\text{'}}\left(m\right)dm \\ ={\left[f\right(m\left)\right]}_0^4 \\ =\mathit{f}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\mathbf{-}\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)} \end{gathered}$$

Therefore, option (A) is the correct answer.