Question

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Answer 1

Given Expression :-

$\sf{f(x)}$

To Integrate :-

The given expression with upper limit as b and lower limit as a.

Solution :-

$\displaystyle{\sf{\int f(x)dx}}$

Applying linearity :-

$\displaystyle{\sf{f \int x\ dx}}$

Applying power rule :-

$\displaystyle{\sf{f \int x^2dx=f \dfrac{x^{1+1}}{1+1} }}\\\\\displaystyle{\sf{=f \dfrac{x^2}{2} }}\\\\\displaystyle{\sf{=\frac{fx^2}{2}+C }}$

Putting "b" in place of x and putting "a" in the place of x and subtracting :-

$\displaystyle{\sf{\frac{f(b)^2}{2}-\frac{f(a)^2}{2} }}\\\\\displaystyle{\sf{=\frac{f(b^2-a^2)}{2} }}\\\\\large\boxed{\boxed{\displaystyle{\sf{ \color{lime}{=\frac{(b-a)(b+a)f}{2} }}}}}$

Answer 2

Step-by-step explanation:

Given :-

• f(x)

Solution :-

=> § f(x) dx

• = F(x) +C .

There ,

F(x) is the function such as F' (x) = f(x)

Extra knowledge :-

• Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral

$\bf \: {{\displaystyle \int _{a}^{b}f(x)\,dx} }$

• can be interpreted informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.