Q. 19 - Calculus

Question

Are definite integrals the “inverse” of differentiation? In other words, does one undo the other? Simplify each of the following to answer this question:

$$\begin{gathered} \left(a\right) {\int }_a^b{f}^{\text{'}}\left(x\right)dx \\ \left(b\right) \frac{d}{dx}{\int }_a^{b}f\left(x\right)dx \end{gathered}$$

Step-by-Step Solution

Answer

Part $\left(a\right)$ The simplified expression for ${\int }_a^b{f}^{\text{'}}\left(x\right)dx$ is, $f\left(b\right)-f\left(a\right)$.

Part $\left(b\right)$ The simplified expression for  $\frac{d}{dx}{\int }_a^{b}f\left(x\right)dx$ is, $0$.

Part a Step 1 . Given information

${\int }_a^b{f}^{\text{'}}\left(x\right)dx$.

Part a Step 2 . The objective is, we need to simplify the given expression.

$\frac{d}{dx}f\left(x\right)=f^{\text{'}}\left(x\right)$

So,

$$\begin{gathered} \int f^{\text{'}}\left(x\right)dx=\left[f\right(x){]}_a^b \\ =f\left(b\right)-f\left(a\right) \end{gathered}$$

Part b Step 1 . Given information

$\frac{d}{dx}{\int }_a^{b}f\left(x\right)dx$.

Part b Step 2 . The objective is, we need to simplify the given expression.

$F\left(x\right)=f\left(x\right)$        [$F\left(x\right)$ is an antiderivative of $f\left(x\right)$]

So,

$$\begin{gathered} \frac{d}{dx}{\int }_a^{b}f\left(x\right)dx=\frac{d}{dx}\left[F\right(x){]}_a^b \\ =\frac{d}{dx}[F\left(b\right)-F\left(a\right)] \end{gathered}$$

                        $=0$           [ $F\left(b\right)-F\left(a\right)$ is a constant and derivative of a constant is $0$]