Question

If $\frac{d}{dx}(ɸ(x\left)\right)=f\left(x\right)$, then ${\int }_1^{2}f\left(x\right)dx=$

• A. $f\left(1\right)–f\left(2\right)$
• B. $\varphi \left(1\right)–\varphi \left(2\right)$
• C. $f\left(2\right)–f\left(1\right)$
• D. $\varphi \left(2\right)–\varphi \left(1\right)$

Answer 1

The correct option is D

$\varphi \left(2\right)–\varphi \left(1\right)$

Find the value of ${\mathbf{\int }}_{\mathbf{1}}^{\mathbf{2}}\mathit{f}\mathbf{}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{}\mathit{d}\mathit{x}$:

Given, $\frac{d}{dx}(ɸ(x\left)\right)=f\left(x\right)$

$\Rightarrow$ ${\int }_1^{2}f\left(x\right)dx=\int \frac{d}{dx}\left(\varphi \left(x\right)\right)dx$

$\Rightarrow$ ${\int }_1^{2}f\left(x\right)dx={\left[\varphi \left(x\right)\right]}_1^2$

$\mathbf{\therefore }{\mathbf{\int }}_{\mathbf{1}}^{\mathbf{2}}\mathit{f}\mathbf{}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{}\mathit{d}\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathit{\varphi }\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{}\mathbf{-}\mathbf{}\mathit{\varphi }\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

Hence, Option ‘D’ is Correct.