Q. 18 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

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

Part a Step 1 . Given information

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

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

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

So,

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

Part b Step 1 . Given information

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

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

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

So,

$$\begin{gathered} \frac{d}{dx}\int f\left(x\right)dx=\frac{d}{dx}F\left(x\right) \\ =f\left(x\right) \end{gathered}$$