Question

If $\frac{d}{dx}f\left(x\right)=g\left(x\right)$, then $\underset{a}{\stackrel{b}{\int }}f\left(x\right)g\left(x\right)dx$ =

• A. $\frac{f\left(b\right)-f\left(a\right)}{2}$
• B. $\frac{f\left(a\right)-f\left(b\right)}{2}$
• C. $\frac{f^2\left(b\right)-f^2\left(a\right)}{2}$
• D. $\frac{f^2\left(a\right)-f^2\left(b\right)}{2}$

Answer 1

Answer: (C)

$\frac{f^2\left(b\right)-f^2\left(a\right)}{2}$

${\int }_a^{b}f\left(x\right)g\left(x\right)dx=?$

Let $f\left(x\right)=t$

Then,

$\frac{d}{dx}f\left(x\right)=\frac{dt}{dx}$

$g\left(x\right)=\frac{dt}{dx}$

$g\left(x\right)dx=dt$

Now,

${\int }_a^{b}f\left(x\right)g\left(x\right)dx={\int }_a^{b}tdt$

$={\left[\frac{t^2}{2}\right]}_a^b$

$={\left[\frac{f^2\left(x\right)}{2}\right]}_a^b$

$=\frac{f^2\left(b\right)-f^2\left(a\right)}{2}$